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Theorem List for Metamath Proof Explorer - 10501-10600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremac3 10501* Axiom of Choice using abbreviations. The logical equivalence to ax-ac 10498 can be established by chaining aceq0 10157 and aceq2 10158. A standard textbook version of AC is derived from this one in dfac2a 10168, and this version of AC is derived from the textbook version in dfac2b 10169, showing their logical equivalence (see dfac2 10170).

The following sketch will help you understand this version of the axiom. Given any set 𝑥, the axiom says that there exists a 𝑦 that is a collection of unordered pairs, one pair for each nonempty member of 𝑥. One entry in the pair is the member of 𝑥, and the other entry is some arbitrary member of that member of 𝑥. Using the Axiom of Regularity, we can show that 𝑦 is really a set of ordered pairs, very similar to the ordered pair construction opthreg 9657. The key theorem for this (used in the proof of dfac2b 10169) is preleq 9655. With this modified definition of ordered pair, it can be seen that 𝑦 is actually a choice function on the members of 𝑥.

For example, suppose 𝑥 = {{1, 2}, {1, 3}, {2, 3, 4}}. Let us try 𝑦 = {{{1, 2}, 1}, {{1, 3}, 1}, {{2, 3, 4}, 2}}. For the member (of 𝑥) 𝑧 = {1, 2}, the only assignment to 𝑤 and 𝑣 that satisfies the axiom is 𝑤 = 1 and 𝑣 = {{1, 2}, 1}, so there is exactly one 𝑤 as required. We verify the other two members of 𝑥 similarly. Thus, 𝑦 satisfies the axiom. Using our modified ordered pair definition, we can say that 𝑦 corresponds to the choice function {⟨{1, 2}, 1⟩, ⟨{1, 3}, 1⟩, ⟨{2, 3, 4}, 2⟩}. Of course other choices for 𝑦 will also satisfy the axiom, for example 𝑦 = {{{1, 2}, 2}, {{1, 3}, 1}, {{2, 3, 4}, 4}}. What AC tells us is that there exists at least one such 𝑦, but it doesn't tell us which one.

(New usage is discouraged.) (Contributed by NM, 19-Jul-1996.)

𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣))
 
Axiomax-ac2 10502* In order to avoid uses of ax-reg 9631 for derivation of AC equivalents, we provide ax-ac2 10502, which is equivalent to the standard AC of textbooks. This appears to be the shortest known equivalent to the standard AC when expressed in terms of set theory primitives. It was found by Kurt Maes as Theorem ackm 10504. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1789 available. The derivation of ax-ac2 10502 from ax-ac 10498 is shown by Theorem axac2 10505, and the reverse derivation by axac 10506. Note that we use ax-reg 9631 to derive ax-ac 10498 from ax-ac2 10502, but not to derive ax-ac2 10502 from ax-ac 10498. (Contributed by NM, 19-Dec-2016.)
𝑦𝑧𝑣𝑢((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣)))))
 
Theoremaxac3 10503 This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 10502 as an axiom. (Contributed by Mario Carneiro, 6-May-2015.) (Revised by NM, 20-Dec-2016.) (Proof modification is discouraged.)
CHOICE
 
Theoremackm 10504* A remarkable equivalent to the Axiom of Choice that has only five quantifiers (when expanded to use only the primitive predicates = and and in prenex normal form), discovered and proved by Kurt Maes. This establishes a new record, reducing from 6 to 5 the largest number of quantified variables needed by any ZFC axiom. The ZF-equivalence to AC is shown by Theorem dfackm 10205. Maes found this version of AC in April 2004 (replacing a longer version, also with five quantifiers, that he found in November 2003). See Kurt Maes, "A 5-quantifier (∈ , =)-expression ZF-equivalent to the Axiom of Choice", https://doi.org/10.48550/arXiv.0705.3162 10205.

The original FOM posts are: http://www.cs.nyu.edu/pipermail/fom/2003-November/007631.html 10205 http://www.cs.nyu.edu/pipermail/fom/2003-November/007641.html 10205. (Contributed by NM, 29-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.) (Proof modification is discouraged.)

𝑥𝑦𝑧𝑣𝑢((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣)))))
 
Theoremaxac2 10505* Derive ax-ac2 10502 from ax-ac 10498. (Contributed by NM, 19-Dec-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑦𝑧𝑣𝑢((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣)))))
 
Theoremaxac 10506* Derive ax-ac 10498 from ax-ac2 10502. Note that ax-reg 9631 is used by the proof. (Contributed by NM, 19-Dec-2016.) (Proof modification is discouraged.)
𝑦𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣))
 
Theoremaxaci 10507 Apply a choice equivalent. (Contributed by Mario Carneiro, 17-May-2015.)
(CHOICE ↔ ∀𝑥𝜑)       𝜑
 
Theoremcardeqv 10508 All sets are well-orderable under choice. (Contributed by Mario Carneiro, 28-Apr-2015.)
dom card = V
 
Theoremnumth3 10509 All sets are well-orderable under choice. (Contributed by Stefan O'Rear, 28-Feb-2015.)
(𝐴𝑉𝐴 ∈ dom card)
 
Theoremnumth2 10510* Numeration theorem: any set is equinumerous to some ordinal (using AC). Theorem 10.3 of [TakeutiZaring] p. 84. (Contributed by NM, 20-Oct-2003.)
𝐴 ∈ V       𝑥 ∈ On 𝑥𝐴
 
Theoremnumth 10511* Numeration theorem: every set can be put into one-to-one correspondence with some ordinal (using AC). Theorem 10.3 of [TakeutiZaring] p. 84. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Mario Carneiro, 8-Jan-2015.)
𝐴 ∈ V       𝑥 ∈ On ∃𝑓 𝑓:𝑥1-1-onto𝐴
 
Theoremac7 10512* An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49. (Contributed by NM, 29-Apr-2004.)
𝑓(𝑓𝑥𝑓 Fn dom 𝑥)
 
Theoremac7g 10513* An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49. (Contributed by NM, 23-Jul-2004.)
(𝑅𝐴 → ∃𝑓(𝑓𝑅𝑓 Fn dom 𝑅))
 
Theoremac4 10514* Equivalent of Axiom of Choice. We do not insist that 𝑓 be a function. However, Theorem ac5 10516, derived from this one, shows that this form of the axiom does imply that at least one such set 𝑓 whose existence we assert is in fact a function. Axiom of Choice of [TakeutiZaring] p. 83.

Takeuti and Zaring call this "weak choice" in contrast to "strong choice" 𝐹𝑧(𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧), which asserts the existence of a universal choice function but requires second-order quantification on (proper) class variable 𝐹 and thus cannot be expressed in our first-order formalization. However, it has been shown that ZF plus strong choice is a conservative extension of ZF plus weak choice. See Ulrich Felgner, "Comparison of the axioms of local and universal choice", Fundamenta Mathematica, 71, 43-62 (1971).

Weak choice can be strengthened in a different direction to choose from a collection of proper classes; see ac6s5 10530. (Contributed by NM, 21-Jul-1996.)

𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)
 
Theoremac4c 10515* Equivalent of Axiom of Choice (class version). (Contributed by NM, 10-Feb-1997.)
𝐴 ∈ V       𝑓𝑥𝐴 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥)
 
Theoremac5 10516* An Axiom of Choice equivalent: there exists a function 𝑓 (called a choice function) with domain 𝐴 that maps each nonempty member of the domain to an element of that member. Axiom AC of [BellMachover] p. 488. Note that the assertion that 𝑓 be a function is not necessary; see ac4 10514. (Contributed by NM, 29-Aug-1999.)
𝐴 ∈ V       𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥))
 
Theoremac5b 10517* Equivalent of Axiom of Choice. (Contributed by NM, 31-Aug-1999.)
𝐴 ∈ V       (∀𝑥𝐴 𝑥 ≠ ∅ → ∃𝑓(𝑓:𝐴 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥))
 
Theoremac6num 10518* A version of ac6 10519 which takes the choice as a hypothesis. (Contributed by Mario Carneiro, 27-Aug-2015.)
(𝑦 = (𝑓𝑥) → (𝜑𝜓))       ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))
 
Theoremac6 10519* Equivalent of Axiom of Choice. This is useful for proving that there exists, for example, a sequence mapping natural numbers to members of a larger set 𝐵, where 𝜑 depends on 𝑥 (the natural number) and 𝑦 (to specify a member of 𝐵). A stronger version of this theorem, ac6s 10523, allows 𝐵 to be a proper class. (Contributed by NM, 18-Oct-1999.) (Revised by Mario Carneiro, 27-Aug-2015.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))
 
Theoremac6c4 10520* Equivalent of Axiom of Choice. 𝐵 is a collection 𝐵(𝑥) of nonempty sets. (Contributed by Mario Carneiro, 22-Mar-2013.)
𝐴 ∈ V    &   𝐵 ∈ V       (∀𝑥𝐴 𝐵 ≠ ∅ → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
 
Theoremac6c5 10521* Equivalent of Axiom of Choice. 𝐵 is a collection 𝐵(𝑥) of nonempty sets. Remark after Theorem 10.46 of [TakeutiZaring] p. 98. (Contributed by Mario Carneiro, 22-Mar-2013.)
𝐴 ∈ V    &   𝐵 ∈ V       (∀𝑥𝐴 𝐵 ≠ ∅ → ∃𝑓𝑥𝐴 (𝑓𝑥) ∈ 𝐵)
 
Theoremac9 10522* An Axiom of Choice equivalent: the infinite Cartesian product of nonempty classes is nonempty. Axiom of Choice (second form) of [Enderton] p. 55 and its converse. (Contributed by Mario Carneiro, 22-Mar-2013.)
𝐴 ∈ V    &   𝐵 ∈ V       (∀𝑥𝐴 𝐵 ≠ ∅ ↔ X𝑥𝐴 𝐵 ≠ ∅)
 
Theoremac6s 10523* Equivalent of Axiom of Choice. Using the Boundedness Axiom bnd2 9932, we derive this strong version of ac6 10519 that doesn't require 𝐵 to be a set. (Contributed by NM, 4-Feb-2004.)
𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))
 
Theoremac6n 10524* Equivalent of Axiom of Choice. Contrapositive of ac6s 10523. (Contributed by NM, 10-Jun-2007.)
𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑓(𝑓:𝐴𝐵 → ∃𝑥𝐴 𝜓) → ∃𝑥𝐴𝑦𝐵 𝜑)
 
Theoremac6s2 10525* Generalization of the Axiom of Choice to classes. Slightly strengthened version of ac6s3 10526. (Contributed by NM, 29-Sep-2006.)
𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑥𝐴𝑦𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 𝜓))
 
Theoremac6s3 10526* Generalization of the Axiom of Choice to classes. Theorem 10.46 of [TakeutiZaring] p. 97. (Contributed by NM, 3-Nov-2004.)
𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑥𝐴𝑦𝜑 → ∃𝑓𝑥𝐴 𝜓)
 
Theoremac6sg 10527* ac6s 10523 with sethood as antecedent. (Contributed by FL, 3-Aug-2009.)
(𝑦 = (𝑓𝑥) → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓)))
 
Theoremac6sf 10528* Version of ac6 10519 with bound-variable hypothesis. (Contributed by NM, 2-Mar-2008.)
𝑦𝜓    &   𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))
 
Theoremac6s4 10529* Generalization of the Axiom of Choice to proper classes. 𝐵 is a collection 𝐵(𝑥) of nonempty, possible proper classes. (Contributed by NM, 29-Sep-2006.)
𝐴 ∈ V       (∀𝑥𝐴 𝐵 ≠ ∅ → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
 
Theoremac6s5 10530* Generalization of the Axiom of Choice to proper classes. 𝐵 is a collection 𝐵(𝑥) of nonempty, possible proper classes. Remark after Theorem 10.46 of [TakeutiZaring] p. 98. (Contributed by NM, 27-Mar-2006.)
𝐴 ∈ V       (∀𝑥𝐴 𝐵 ≠ ∅ → ∃𝑓𝑥𝐴 (𝑓𝑥) ∈ 𝐵)
 
Theoremac8 10531* An Axiom of Choice equivalent. Given a family 𝑥 of mutually disjoint nonempty sets, there exists a set 𝑦 containing exactly one member from each set in the family. Theorem 6M(4) of [Enderton] p. 151. (Contributed by NM, 14-May-2004.)
((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦))
 
Theoremac9s 10532* An Axiom of Choice equivalent: the infinite Cartesian product of nonempty classes is nonempty. Axiom of Choice (second form) of [Enderton] p. 55 and its converse. This is a stronger version of the axiom in Enderton, with no existence requirement for the family of classes 𝐵(𝑥) (achieved via the Collection Principle cp 9930). (Contributed by NM, 29-Sep-2006.)
𝐴 ∈ V       (∀𝑥𝐴 𝐵 ≠ ∅ ↔ X𝑥𝐴 𝐵 ≠ ∅)
 
3.2.2  AC equivalents: well-ordering, Zorn's lemma
 
Theoremnumthcor 10533* Any set is strictly dominated by some ordinal. (Contributed by NM, 22-Oct-2003.)
(𝐴𝑉 → ∃𝑥 ∈ On 𝐴𝑥)
 
Theoremweth 10534* Well-ordering theorem: any set 𝐴 can be well-ordered. This is an equivalent of the Axiom of Choice. Theorem 6 of [Suppes] p. 242. First proved by Ernst Zermelo (the "Z" in ZFC) in 1904. (Contributed by Mario Carneiro, 5-Jan-2013.)
(𝐴𝑉 → ∃𝑥 𝑥 We 𝐴)
 
Theoremzorn2lem1 10535* Lemma for zorn2 10545. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))    &   𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}       ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → (𝐹𝑥) ∈ 𝐷)
 
Theoremzorn2lem2 10536* Lemma for zorn2 10545. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))    &   𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}       ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → (𝑦𝑥 → (𝐹𝑦)𝑅(𝐹𝑥)))
 
Theoremzorn2lem3 10537* Lemma for zorn2 10545. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))    &   𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}       ((𝑅 Po 𝐴 ∧ (𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅))) → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))
 
Theoremzorn2lem4 10538* Lemma for zorn2 10545. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))    &   𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}       ((𝑅 Po 𝐴𝑤 We 𝐴) → ∃𝑥 ∈ On 𝐷 = ∅)
 
Theoremzorn2lem5 10539* Lemma for zorn2 10545. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))    &   𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}    &   𝐻 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑦)𝑔𝑅𝑧}       (((𝑤 We 𝐴𝑥 ∈ On) ∧ ∀𝑦𝑥 𝐻 ≠ ∅) → (𝐹𝑥) ⊆ 𝐴)
 
Theoremzorn2lem6 10540* Lemma for zorn2 10545. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))    &   𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}    &   𝐻 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑦)𝑔𝑅𝑧}       (𝑅 Po 𝐴 → (((𝑤 We 𝐴𝑥 ∈ On) ∧ ∀𝑦𝑥 𝐻 ≠ ∅) → 𝑅 Or (𝐹𝑥)))
 
Theoremzorn2lem7 10541* Lemma for zorn2 10545. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))    &   𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}    &   𝐻 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑦)𝑔𝑅𝑧}       ((𝐴 ∈ dom card ∧ 𝑅 Po 𝐴 ∧ ∀𝑠((𝑠𝐴𝑅 Or 𝑠) → ∃𝑎𝐴𝑟𝑠 (𝑟𝑅𝑎𝑟 = 𝑎))) → ∃𝑎𝐴𝑏𝐴 ¬ 𝑎𝑅𝑏)
 
Theoremzorn2g 10542* Zorn's Lemma of [Monk1] p. 117. This version of zorn2 10545 avoids the Axiom of Choice by assuming that 𝐴 is well-orderable. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
((𝐴 ∈ dom card ∧ 𝑅 Po 𝐴 ∧ ∀𝑤((𝑤𝐴𝑅 Or 𝑤) → ∃𝑥𝐴𝑧𝑤 (𝑧𝑅𝑥𝑧 = 𝑥))) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑅𝑦)
 
Theoremzorng 10543* Zorn's Lemma. If the union of every chain (with respect to inclusion) in a set belongs to the set, then the set contains a maximal element. Theorem 6M of [Enderton] p. 151. This version of zorn 10546 avoids the Axiom of Choice by assuming that 𝐴 is well-orderable. (Contributed by NM, 12-Aug-2004.) (Revised by Mario Carneiro, 9-May-2015.)
((𝐴 ∈ dom card ∧ ∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → 𝑧𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
 
Theoremzornn0g 10544* Variant of Zorn's lemma zorng 10543 in which , the union of the empty chain, is not required to be an element of 𝐴. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Mario Carneiro, 9-May-2015.)
((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
 
Theoremzorn2 10545* Zorn's Lemma of [Monk1] p. 117. This theorem is equivalent to the Axiom of Choice and states that every partially ordered set 𝐴 (with an ordering relation 𝑅) in which every totally ordered subset has an upper bound, contains at least one maximal element. The main proof consists of lemmas zorn2lem1 10535 through zorn2lem7 10541; this final piece mainly changes bound variables to eliminate the hypotheses of zorn2lem7 10541. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐴 ∈ V       ((𝑅 Po 𝐴 ∧ ∀𝑤((𝑤𝐴𝑅 Or 𝑤) → ∃𝑥𝐴𝑧𝑤 (𝑧𝑅𝑥𝑧 = 𝑥))) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑅𝑦)
 
Theoremzorn 10546* Zorn's Lemma. If the union of every chain (with respect to inclusion) in a set belongs to the set, then the set contains a maximal element. This theorem is equivalent to the Axiom of Choice. Theorem 6M of [Enderton] p. 151. See zorn2 10545 for a version with general partial orderings. (Contributed by NM, 12-Aug-2004.)
𝐴 ∈ V       (∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → 𝑧𝐴) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
 
Theoremzornn0 10547* Variant of Zorn's lemma zorn 10546 in which , the union of the empty chain, is not required to be an element of 𝐴. (Contributed by Jeff Madsen, 5-Jan-2011.)
𝐴 ∈ V       ((𝐴 ≠ ∅ ∧ ∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
 
Theoremttukeylem1 10548* Lemma for ttukey 10557. Expand out the property of being an element of a property of finite character. (Contributed by Mario Carneiro, 15-May-2015.)
(𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))       (𝜑 → (𝐶𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))
 
Theoremttukeylem2 10549* Lemma for ttukey 10557. A property of finite character is closed under subsets. (Contributed by Mario Carneiro, 15-May-2015.)
(𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))       ((𝜑 ∧ (𝐶𝐴𝐷𝐶)) → 𝐷𝐴)
 
Theoremttukeylem3 10550* Lemma for ttukey 10557. (Contributed by Mario Carneiro, 11-May-2015.)
(𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))    &   𝐺 = recs((𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅)))))       ((𝜑𝐶 ∈ On) → (𝐺𝐶) = if(𝐶 = 𝐶, if(𝐶 = ∅, 𝐵, (𝐺𝐶)), ((𝐺 𝐶) ∪ if(((𝐺 𝐶) ∪ {(𝐹 𝐶)}) ∈ 𝐴, {(𝐹 𝐶)}, ∅))))
 
Theoremttukeylem4 10551* Lemma for ttukey 10557. (Contributed by Mario Carneiro, 15-May-2015.)
(𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))    &   𝐺 = recs((𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅)))))       (𝜑 → (𝐺‘∅) = 𝐵)
 
Theoremttukeylem5 10552* Lemma for ttukey 10557. The 𝐺 function forms a (transfinitely long) chain of inclusions. (Contributed by Mario Carneiro, 15-May-2015.)
(𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))    &   𝐺 = recs((𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅)))))       ((𝜑 ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On ∧ 𝐶𝐷)) → (𝐺𝐶) ⊆ (𝐺𝐷))
 
Theoremttukeylem6 10553* Lemma for ttukey 10557. (Contributed by Mario Carneiro, 15-May-2015.)
(𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))    &   𝐺 = recs((𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅)))))       ((𝜑𝐶 ∈ suc (card‘( 𝐴𝐵))) → (𝐺𝐶) ∈ 𝐴)
 
Theoremttukeylem7 10554* Lemma for ttukey 10557. (Contributed by Mario Carneiro, 15-May-2015.)
(𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))    &   𝐺 = recs((𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅)))))       (𝜑 → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦))
 
Theoremttukey2g 10555* The Teichmüller-Tukey Lemma ttukey 10557 with a slightly stronger conclusion: we can set up the maximal element of 𝐴 so that it also contains some given 𝐵𝐴 as a subset. (Contributed by Mario Carneiro, 15-May-2015.)
(( 𝐴 ∈ dom card ∧ 𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦))
 
Theoremttukeyg 10556* The Teichmüller-Tukey Lemma ttukey 10557 stated with the "choice" as an antecedent (the hypothesis 𝐴 ∈ dom card says that 𝐴 is well-orderable). (Contributed by Mario Carneiro, 15-May-2015.)
(( 𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
 
Theoremttukey 10557* The Teichmüller-Tukey Lemma, an Axiom of Choice equivalent. If 𝐴 is a nonempty collection of finite character, then 𝐴 has a maximal element with respect to inclusion. Here "finite character" means that 𝑥𝐴 iff every finite subset of 𝑥 is in 𝐴. (Contributed by Mario Carneiro, 15-May-2015.)
𝐴 ∈ V       ((𝐴 ≠ ∅ ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
 
Theoremaxdclem 10558* Lemma for axdc 10560. (Contributed by Mario Carneiro, 25-Jan-2013.)
𝐹 = (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧𝑦𝑥𝑧})), 𝑠) ↾ ω)       ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥 ∧ ∃𝑧(𝐹𝐾)𝑥𝑧) → (𝐾 ∈ ω → (𝐹𝐾)𝑥(𝐹‘suc 𝐾)))
 
Theoremaxdclem2 10559* Lemma for axdc 10560. Using the full Axiom of Choice, we can construct a choice function 𝑔 on 𝒫 dom 𝑥. From this, we can build a sequence 𝐹 starting at any value 𝑠 ∈ dom 𝑥 by repeatedly applying 𝑔 to the set (𝐹𝑥) (where 𝑥 is the value from the previous iteration). (Contributed by Mario Carneiro, 25-Jan-2013.)
𝐹 = (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧𝑦𝑥𝑧})), 𝑠) ↾ ω)       (∃𝑧 𝑠𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)))
 
Theoremaxdc 10560* This theorem derives ax-dc 10485 using ax-ac 10498 and ax-inf 9677. Thus, AC implies DC, but not vice-versa (so that ZFC is strictly stronger than ZF+DC). (New usage is discouraged.) (Contributed by Mario Carneiro, 25-Jan-2013.)
((∃𝑦𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛))
 
Theoremfodomg 10561 An onto function implies dominance of domain over range. Lemma 10.20 of [Kunen] p. 30. This theorem uses the axiom of choice ac7g 10513. The axiom of choice is not needed for finite sets, see fodomfi 9365. See also fodomnum 10096. (Contributed by NM, 23-Jul-2004.) (Proof shortened by BJ, 20-May-2024.)
(𝐴𝑉 → (𝐹:𝐴onto𝐵𝐵𝐴))
 
Theoremfodom 10562 An onto function implies dominance of domain over range. (Contributed by NM, 23-Jul-2004.)
𝐴 ∈ V       (𝐹:𝐴onto𝐵𝐵𝐴)
 
Theoremdmct 10563 The domain of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
(𝐴 ≼ ω → dom 𝐴 ≼ ω)
 
Theoremrnct 10564 The range of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
(𝐴 ≼ ω → ran 𝐴 ≼ ω)
 
Theoremfodomb 10565* Equivalence of an onto mapping and dominance for a nonempty set. Proposition 10.35 of [TakeutiZaring] p. 93. (Contributed by NM, 29-Jul-2004.)
((𝐴 ≠ ∅ ∧ ∃𝑓 𝑓:𝐴onto𝐵) ↔ (∅ ≺ 𝐵𝐵𝐴))
 
Theoremwdomac 10566 When assuming AC, weak and usual dominance coincide. It is not known if this is an AC equivalent. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
(𝑋* 𝑌𝑋𝑌)
 
Theorembrdom3 10567* Equivalence to a dominance relation. (Contributed by NM, 27-Mar-2007.)
𝐵 ∈ V       (𝐴𝐵 ↔ ∃𝑓(∀𝑥∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥))
 
Theorembrdom5 10568* An equivalence to a dominance relation. (Contributed by NM, 29-Mar-2007.)
𝐵 ∈ V       (𝐴𝐵 ↔ ∃𝑓(∀𝑥𝐵 ∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥))
 
Theorembrdom4 10569* An equivalence to a dominance relation. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.)
𝐵 ∈ V       (𝐴𝐵 ↔ ∃𝑓(∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥))
 
Theorembrdom7disj 10570* An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 29-Mar-2007.) (Revised by NM, 16-Jun-2017.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝐴𝐵) = ∅       (𝐴𝐵 ↔ ∃𝑓(∀𝑥𝐵 ∃*𝑦𝐴 {𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓))
 
Theorembrdom6disj 10571* An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 5-Apr-2007.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝐴𝐵) = ∅       (𝐴𝐵 ↔ ∃𝑓(∀𝑥𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓))
 
Theoremfin71ac 10572 Once we allow AC, the "strongest" definition of finite set becomes equivalent to the "weakest" and the entire hierarchy collapses. (Contributed by Stefan O'Rear, 29-Oct-2014.)
FinVII = Fin
 
Theoremimadomg 10573 An image of a function under a set is dominated by the set. Proposition 10.34 of [TakeutiZaring] p. 92. (Contributed by NM, 23-Jul-2004.)
(𝐴𝐵 → (Fun 𝐹 → (𝐹𝐴) ≼ 𝐴))
 
Theoremfimact 10574 The image by a function of a countable set is countable. (Contributed by Thierry Arnoux, 27-Mar-2018.)
((𝐴 ≼ ω ∧ Fun 𝐹) → (𝐹𝐴) ≼ ω)
 
Theoremfnrndomg 10575 The range of a function is dominated by its domain. (Contributed by NM, 1-Sep-2004.)
(𝐴𝐵 → (𝐹 Fn 𝐴 → ran 𝐹𝐴))
 
Theoremfnct 10576 If the domain of a function is countable, the function is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐹 ≼ ω)
 
Theoremmptct 10577* A countable mapping set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
(𝐴 ≼ ω → (𝑥𝐴𝐵) ≼ ω)
 
Theoremiunfo 10578* Existence of an onto function from a disjoint union to a union. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 18-Jan-2014.)
𝑇 = 𝑥𝐴 ({𝑥} × 𝐵)       (2nd𝑇):𝑇onto 𝑥𝐴 𝐵
 
Theoremiundom2g 10579* An upper bound for the cardinality of a disjoint indexed union, with explicit choice principles. 𝐵 depends on 𝑥 and should be thought of as 𝐵(𝑥). (Contributed by Mario Carneiro, 1-Sep-2015.)
𝑇 = 𝑥𝐴 ({𝑥} × 𝐵)    &   (𝜑 𝑥𝐴 (𝐶m 𝐵) ∈ AC 𝐴)    &   (𝜑 → ∀𝑥𝐴 𝐵𝐶)       (𝜑𝑇 ≼ (𝐴 × 𝐶))
 
Theoremiundomg 10580* An upper bound for the cardinality of an indexed union, with explicit choice principles. 𝐵 depends on 𝑥 and should be thought of as 𝐵(𝑥). (Contributed by Mario Carneiro, 1-Sep-2015.)
𝑇 = 𝑥𝐴 ({𝑥} × 𝐵)    &   (𝜑 𝑥𝐴 (𝐶m 𝐵) ∈ AC 𝐴)    &   (𝜑 → ∀𝑥𝐴 𝐵𝐶)    &   (𝜑 → (𝐴 × 𝐶) ∈ AC 𝑥𝐴 𝐵)       (𝜑 𝑥𝐴 𝐵 ≼ (𝐴 × 𝐶))
 
Theoremiundom 10581* An upper bound for the cardinality of an indexed union. 𝐶 depends on 𝑥 and should be thought of as 𝐶(𝑥). (Contributed by NM, 26-Mar-2006.)
((𝐴𝑉 ∧ ∀𝑥𝐴 𝐶𝐵) → 𝑥𝐴 𝐶 ≼ (𝐴 × 𝐵))
 
Theoremunidom 10582* An upper bound for the cardinality of a union. Theorem 10.47 of [TakeutiZaring] p. 98. (Contributed by NM, 25-Mar-2006.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)
((𝐴𝑉 ∧ ∀𝑥𝐴 𝑥𝐵) → 𝐴 ≼ (𝐴 × 𝐵))
 
Theoremuniimadom 10583* An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. (Contributed by NM, 25-Mar-2006.)
𝐴 ∈ V    &   𝐵 ∈ V       ((Fun 𝐹 ∧ ∀𝑥𝐴 (𝐹𝑥) ≼ 𝐵) → (𝐹𝐴) ≼ (𝐴 × 𝐵))
 
Theoremuniimadomf 10584* An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 10583 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by NM, 26-Mar-2006.)
𝑥𝐹    &   𝐴 ∈ V    &   𝐵 ∈ V       ((Fun 𝐹 ∧ ∀𝑥𝐴 (𝐹𝑥) ≼ 𝐵) → (𝐹𝐴) ≼ (𝐴 × 𝐵))
 
3.2.3  Cardinal number theorems using Axiom of Choice
 
Theoremcardval 10585* The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 10030 for a simpler version of its value. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 28-Apr-2015.)
𝐴 ∈ V       (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴}
 
Theoremcardid 10586 Any set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 28-Apr-2015.)
𝐴 ∈ V       (card‘𝐴) ≈ 𝐴
 
Theoremcardidg 10587 Any set is equinumerous to its cardinal number. Closed theorem form of cardid 10586. (Contributed by David Moews, 1-May-2017.)
(𝐴𝐵 → (card‘𝐴) ≈ 𝐴)
 
Theoremcardidd 10588 Any set is equinumerous to its cardinal number. Deduction form of cardid 10586. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)       (𝜑 → (card‘𝐴) ≈ 𝐴)
 
Theoremcardf 10589 The cardinality function is a function with domain the well-orderable sets. Assuming AC, this is the universe. (Contributed by Mario Carneiro, 6-Jun-2013.) (Revised by Mario Carneiro, 13-Sep-2013.)
card:V⟶On
 
Theoremcarden 10590 Two sets are equinumerous iff their cardinal numbers are equal. This important theorem expresses the essential concept behind "cardinality" or "size". This theorem appears as Proposition 10.10 of [TakeutiZaring] p. 85, Theorem 7P of [Enderton] p. 197, and Theorem 9 of [Suppes] p. 242 (among others). The Axiom of Choice is required for its proof. Related theorems are hasheni 14360 and the finite-set-only hashen 14359.

This theorem is also known as Hume's Principle. Gottlob Frege's two-volume Grundgesetze der Arithmetik used his Basic Law V to prove this theorem. Unfortunately Basic Law V caused Frege's system to be inconsistent because it was subject to Russell's paradox (see ru 3773). Later scholars have found that Frege primarily used Basic Law V to Hume's Principle. If Basic Law V is replaced by Hume's Principle in Frege's system, much of Frege's work is restored. Grundgesetze der Arithmetik, once Basic Law V is replaced, proves "Frege's theorem" (the Peano axioms of arithmetic can be derived in second-order logic from Hume's principle). See https://plato.stanford.edu/entries/frege-theorem 3773. We take a different approach, using first-order logic and ZFC, to prove the Peano axioms of arithmetic.

The theory of cardinality can also be developed without AC by introducing "card" as a primitive notion and stating this theorem as an axiom, as is done with the axiom for cardinal numbers in [Suppes] p. 111. Finally, if we allow the Axiom of Regularity, we can avoid AC by defining the cardinal number of a set as the set of all sets equinumerous to it and having the least possible rank (see karden 9934). (Contributed by NM, 22-Oct-2003.)

((𝐴𝐶𝐵𝐷) → ((card‘𝐴) = (card‘𝐵) ↔ 𝐴𝐵))
 
Theoremcardeq0 10591 Only the empty set has cardinality zero. (Contributed by NM, 23-Apr-2004.)
(𝐴𝑉 → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
 
Theoremunsnen 10592 Equinumerosity of a set with a new element added. (Contributed by NM, 7-Nov-2008.)
𝐴 ∈ V    &   𝐵 ∈ V       𝐵𝐴 → (𝐴 ∪ {𝐵}) ≈ suc (card‘𝐴))
 
Theoremcarddom 10593 Two sets have the dominance relationship iff their cardinalities have the subset relationship. Equation i of [Quine] p. 232. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 30-Apr-2015.)
((𝐴𝑉𝐵𝑊) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴𝐵))
 
Theoremcardsdom 10594 Two sets have the strict dominance relationship iff their cardinalities have the membership relationship. Corollary 19.7(2) of [Eisenberg] p. 310. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 30-Apr-2015.)
((𝐴𝑉𝐵𝑊) → ((card‘𝐴) ∈ (card‘𝐵) ↔ 𝐴𝐵))
 
Theoremdomtri 10595 Trichotomy law for dominance and strict dominance. This theorem is equivalent to the Axiom of Choice. (Contributed by NM, 4-Jan-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
 
Theorementric 10596 Trichotomy of equinumerosity and strict dominance. This theorem is equivalent to the Axiom of Choice. Theorem 8 of [Suppes] p. 242. (Contributed by NM, 4-Jan-2004.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵𝐴𝐵𝐵𝐴))
 
Theorementri2 10597 Trichotomy of dominance and strict dominance. (Contributed by NM, 4-Jan-2004.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵𝐵𝐴))
 
Theorementri3 10598 Trichotomy of dominance. This theorem is equivalent to the Axiom of Choice. Part of Proposition 4.42(d) of [Mendelson] p. 275. (Contributed by NM, 4-Jan-2004.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵𝐵𝐴))
 
Theoremsdomsdomcard 10599 A set strictly dominates iff its cardinal strictly dominates. (Contributed by NM, 30-Oct-2003.)
(𝐴𝐵𝐴 ≺ (card‘𝐵))
 
Theoremcanth3 10600 Cantor's theorem in terms of cardinals. This theorem tells us that no matter how large a cardinal number is, there is a still larger cardinal number. Theorem 18.12 of [Monk1] p. 133. (Contributed by NM, 5-Nov-2003.)
(𝐴𝑉 → (card‘𝐴) ∈ (card‘𝒫 𝐴))
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