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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | pw2f1olem 9101* | Lemma for pw2f1o 9102. (Contributed by Mario Carneiro, 6-Oct-2014.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) ⇒ ⊢ (𝜑 → ((𝑆 ∈ 𝒫 𝐴 ∧ 𝐺 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑆, 𝐶, 𝐵))) ↔ (𝐺 ∈ ({𝐵, 𝐶} ↑m 𝐴) ∧ 𝑆 = (◡𝐺 “ {𝐶})))) | ||
Theorem | pw2f1o 9102* | The power set of a set is equinumerous to set exponentiation with an unordered pair base of ordinal 2. Generalized from Proposition 10.44 of [TakeutiZaring] p. 96. (Contributed by Mario Carneiro, 6-Oct-2014.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) & ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, 𝐶, 𝐵))) ⇒ ⊢ (𝜑 → 𝐹:𝒫 𝐴–1-1-onto→({𝐵, 𝐶} ↑m 𝐴)) | ||
Theorem | pw2eng 9103 | The power set of a set is equinumerous to set exponentiation with a base of ordinal 2o. (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 1-Jul-2015.) |
⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≈ (2o ↑m 𝐴)) | ||
Theorem | pw2en 9104 | The power set of a set is equinumerous to set exponentiation with a base of ordinal 2. Proposition 10.44 of [TakeutiZaring] p. 96. This is Metamath 100 proof #52. (Contributed by NM, 29-Jan-2004.) (Proof shortened by Mario Carneiro, 1-Jul-2015.) |
⊢ 𝐴 ∈ V ⇒ ⊢ 𝒫 𝐴 ≈ (2o ↑m 𝐴) | ||
Theorem | fopwdom 9105 | Covering implies injection on power sets. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.) (Revised by AV, 18-Sep-2021.) |
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → 𝒫 𝐵 ≼ 𝒫 𝐴) | ||
Theorem | enfixsn 9106* | Given two equipollent sets, a bijection can always be chosen which fixes a single point. (Contributed by Stefan O'Rear, 9-Jul-2015.) |
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) → ∃𝑓(𝑓:𝑋–1-1-onto→𝑌 ∧ (𝑓‘𝐴) = 𝐵)) | ||
Theorem | sucdom2OLD 9107 | Obsolete version of sucdom2 9231 as of 4-Dec-2024. (Contributed by Mario Carneiro, 12-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 ≺ 𝐵 → suc 𝐴 ≼ 𝐵) | ||
Theorem | sbthlem1 9108* | Lemma for sbth 9118. (Contributed by NM, 22-Mar-1998.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} ⇒ ⊢ ∪ 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) | ||
Theorem | sbthlem2 9109* | Lemma for sbth 9118. (Contributed by NM, 22-Mar-1998.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} ⇒ ⊢ (ran 𝑔 ⊆ 𝐴 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ ∪ 𝐷) | ||
Theorem | sbthlem3 9110* | Lemma for sbth 9118. (Contributed by NM, 22-Mar-1998.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} ⇒ ⊢ (ran 𝑔 ⊆ 𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (𝐴 ∖ ∪ 𝐷)) | ||
Theorem | sbthlem4 9111* | Lemma for sbth 9118. (Contributed by NM, 27-Mar-1998.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} ⇒ ⊢ (((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = (𝐵 ∖ (𝑓 “ ∪ 𝐷))) | ||
Theorem | sbthlem5 9112* | Lemma for sbth 9118. (Contributed by NM, 22-Mar-1998.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} & ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ⇒ ⊢ ((dom 𝑓 = 𝐴 ∧ ran 𝑔 ⊆ 𝐴) → dom 𝐻 = 𝐴) | ||
Theorem | sbthlem6 9113* | Lemma for sbth 9118. (Contributed by NM, 27-Mar-1998.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} & ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ⇒ ⊢ ((ran 𝑓 ⊆ 𝐵 ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → ran 𝐻 = 𝐵) | ||
Theorem | sbthlem7 9114* | Lemma for sbth 9118. (Contributed by NM, 27-Mar-1998.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} & ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ⇒ ⊢ ((Fun 𝑓 ∧ Fun ◡𝑔) → Fun 𝐻) | ||
Theorem | sbthlem8 9115* | Lemma for sbth 9118. (Contributed by NM, 27-Mar-1998.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} & ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ⇒ ⊢ ((Fun ◡𝑓 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → Fun ◡𝐻) | ||
Theorem | sbthlem9 9116* | Lemma for sbth 9118. (Contributed by NM, 28-Mar-1998.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} & ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ⇒ ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝐻:𝐴–1-1-onto→𝐵) | ||
Theorem | sbthlem10 9117* | Lemma for sbth 9118. (Contributed by NM, 28-Mar-1998.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} & ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) & ⊢ 𝐵 ∈ V ⇒ ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵) | ||
Theorem | sbth 9118 |
Schroeder-Bernstein Theorem. Theorem 18 of [Suppes] p. 95. This
theorem states that if set 𝐴 is smaller (has lower cardinality)
than
𝐵 and vice-versa, then 𝐴 and
𝐵
are equinumerous (have the
same cardinality). The interesting thing is that this can be proved
without invoking the Axiom of Choice, as we do here. The theorem can
also be proved from the axiom of choice and the linear order of the
cardinal numbers, but our development does not provide the linear order
of cardinal numbers until much later and in ways that depend on
Schroeder-Bernstein.
The main proof consists of lemmas sbthlem1 9108 through sbthlem10 9117; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlem10 9117. We follow closely the proof in Suppes, which you should consult to understand our proof at a higher level. Note that Suppes' proof, which is credited to J. M. Whitaker, does not require the Axiom of Infinity. In the Intuitionistic Logic Explorer (ILE) the Schroeder-Bernstein Theorem has been proven equivalent to the law of the excluded middle (LEM), and in ILE the LEM is not accepted as necessarily true; see https://us.metamath.org/ileuni/exmidsbth.html 9117. This is Metamath 100 proof #25. (Contributed by NM, 8-Jun-1998.) |
⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵) | ||
Theorem | sbthb 9119 | Schroeder-Bernstein Theorem and its converse. (Contributed by NM, 8-Jun-1998.) |
⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) ↔ 𝐴 ≈ 𝐵) | ||
Theorem | sbthcl 9120 | Schroeder-Bernstein Theorem in class form. (Contributed by NM, 28-Mar-1998.) |
⊢ ≈ = ( ≼ ∩ ◡ ≼ ) | ||
Theorem | dfsdom2 9121 | Alternate definition of strict dominance. Compare Definition 3 of [Suppes] p. 97. (Contributed by NM, 31-Mar-1998.) |
⊢ ≺ = ( ≼ ∖ ◡ ≼ ) | ||
Theorem | brsdom2 9122 | Alternate definition of strict dominance. Definition 3 of [Suppes] p. 97. (Contributed by NM, 27-Jul-2004.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ≺ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐵 ≼ 𝐴)) | ||
Theorem | sdomnsym 9123 | Strict dominance is asymmetric. Theorem 21(ii) of [Suppes] p. 97. (Contributed by NM, 8-Jun-1998.) |
⊢ (𝐴 ≺ 𝐵 → ¬ 𝐵 ≺ 𝐴) | ||
Theorem | domnsym 9124 | Theorem 22(i) of [Suppes] p. 97. (Contributed by NM, 10-Jun-1998.) |
⊢ (𝐴 ≼ 𝐵 → ¬ 𝐵 ≺ 𝐴) | ||
Theorem | 0domg 9125 | Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) Avoid ax-pow 5365, ax-un 7741. (Revised by BTernaryTau, 29-Nov-2024.) |
⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴) | ||
Theorem | 0domgOLD 9126 | Obsolete version of 0domg 9125 as of 29-Nov-2024. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴) | ||
Theorem | dom0 9127 | A set dominated by the empty set is empty. (Contributed by NM, 22-Nov-2004.) Avoid ax-pow 5365, ax-un 7741. (Revised by BTernaryTau, 29-Nov-2024.) |
⊢ (𝐴 ≼ ∅ ↔ 𝐴 = ∅) | ||
Theorem | dom0OLD 9128 | Obsolete version of dom0 9127 as of 29-Nov-2024. (Contributed by NM, 22-Nov-2004.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 ≼ ∅ ↔ 𝐴 = ∅) | ||
Theorem | 0sdomg 9129 | A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 23-Mar-2006.) Avoid ax-pow 5365, ax-un 7741. (Revised by BTernaryTau, 29-Nov-2024.) |
⊢ (𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) | ||
Theorem | 0sdomgOLD 9130 | Obsolete version of 0sdomg 9129 as of 29-Nov-2024. (Contributed by NM, 23-Mar-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)) | ||
Theorem | 0dom 9131 | Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ∅ ≼ 𝐴 | ||
Theorem | 0sdom 9132 | A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 29-Jul-2004.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅) | ||
Theorem | sdom0 9133 | The empty set does not strictly dominate any set. (Contributed by NM, 26-Oct-2003.) Avoid ax-pow 5365, ax-un 7741. (Revised by BTernaryTau, 29-Nov-2024.) |
⊢ ¬ 𝐴 ≺ ∅ | ||
Theorem | sdom0OLD 9134 | Obsolete version of sdom0 9133 as of 29-Nov-2024. (Contributed by NM, 26-Oct-2003.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ¬ 𝐴 ≺ ∅ | ||
Theorem | sdomdomtr 9135 | Transitivity of strict dominance and dominance. Theorem 22(iii) of [Suppes] p. 97. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≺ 𝐶) | ||
Theorem | sdomentr 9136 | Transitivity of strict dominance and equinumerosity. Exercise 11 of [Suppes] p. 98. (Contributed by NM, 26-Oct-2003.) |
⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≺ 𝐶) | ||
Theorem | domsdomtr 9137 | Transitivity of dominance and strict dominance. Theorem 22(ii) of [Suppes] p. 97. (Contributed by NM, 10-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≺ 𝐶) | ||
Theorem | ensdomtr 9138 | Transitivity of equinumerosity and strict dominance. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≺ 𝐶) | ||
Theorem | sdomirr 9139 | Strict dominance is irreflexive. Theorem 21(i) of [Suppes] p. 97. (Contributed by NM, 4-Jun-1998.) |
⊢ ¬ 𝐴 ≺ 𝐴 | ||
Theorem | sdomtr 9140 | Strict dominance is transitive. Theorem 21(iii) of [Suppes] p. 97. (Contributed by NM, 9-Jun-1998.) |
⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≺ 𝐶) | ||
Theorem | sdomn2lp 9141 | Strict dominance has no 2-cycle loops. (Contributed by NM, 6-May-2008.) |
⊢ ¬ (𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝐴) | ||
Theorem | enen1 9142 | Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.) |
⊢ (𝐴 ≈ 𝐵 → (𝐴 ≈ 𝐶 ↔ 𝐵 ≈ 𝐶)) | ||
Theorem | enen2 9143 | Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.) |
⊢ (𝐴 ≈ 𝐵 → (𝐶 ≈ 𝐴 ↔ 𝐶 ≈ 𝐵)) | ||
Theorem | domen1 9144 | Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.) |
⊢ (𝐴 ≈ 𝐵 → (𝐴 ≼ 𝐶 ↔ 𝐵 ≼ 𝐶)) | ||
Theorem | domen2 9145 | Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.) |
⊢ (𝐴 ≈ 𝐵 → (𝐶 ≼ 𝐴 ↔ 𝐶 ≼ 𝐵)) | ||
Theorem | sdomen1 9146 | Equality-like theorem for equinumerosity and strict dominance. (Contributed by NM, 8-Nov-2003.) |
⊢ (𝐴 ≈ 𝐵 → (𝐴 ≺ 𝐶 ↔ 𝐵 ≺ 𝐶)) | ||
Theorem | sdomen2 9147 | Equality-like theorem for equinumerosity and strict dominance. (Contributed by NM, 8-Nov-2003.) |
⊢ (𝐴 ≈ 𝐵 → (𝐶 ≺ 𝐴 ↔ 𝐶 ≺ 𝐵)) | ||
Theorem | domtriord 9148 | Dominance is trichotomous in the restricted case of ordinal numbers. (Contributed by Jeff Hankins, 24-Oct-2009.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴)) | ||
Theorem | sdomel 9149 | For ordinals, strict dominance implies membership. (Contributed by Mario Carneiro, 13-Jan-2013.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ≺ 𝐵 → 𝐴 ∈ 𝐵)) | ||
Theorem | sdomdif 9150 | The difference of a set from a smaller set cannot be empty. (Contributed by Mario Carneiro, 5-Feb-2013.) |
⊢ (𝐴 ≺ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅) | ||
Theorem | onsdominel 9151 | An ordinal with more elements of some type is larger. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ (𝐴 ∩ 𝐶) ≺ (𝐵 ∩ 𝐶)) → 𝐴 ∈ 𝐵) | ||
Theorem | domunsn 9152 | Dominance over a set with one element added. (Contributed by Mario Carneiro, 18-May-2015.) |
⊢ (𝐴 ≺ 𝐵 → (𝐴 ∪ {𝐶}) ≼ 𝐵) | ||
Theorem | fodomr 9153* | There exists a mapping from a set onto any (nonempty) set that it dominates. (Contributed by NM, 23-Mar-2006.) |
⊢ ((∅ ≺ 𝐵 ∧ 𝐵 ≼ 𝐴) → ∃𝑓 𝑓:𝐴–onto→𝐵) | ||
Theorem | pwdom 9154 | Injection of sets implies injection on power sets. (Contributed by Mario Carneiro, 9-Apr-2015.) |
⊢ (𝐴 ≼ 𝐵 → 𝒫 𝐴 ≼ 𝒫 𝐵) | ||
Theorem | canth2 9155 | Cantor's Theorem. No set is equinumerous to its power set. Specifically, any set has a cardinality (size) strictly less than the cardinality of its power set. For example, the cardinality of real numbers is the same as the cardinality of the power set of integers, so real numbers cannot be put into a one-to-one correspondence with integers. Theorem 23 of [Suppes] p. 97. For the function version, see canth 7372. This is Metamath 100 proof #63. (Contributed by NM, 7-Aug-1994.) |
⊢ 𝐴 ∈ V ⇒ ⊢ 𝐴 ≺ 𝒫 𝐴 | ||
Theorem | canth2g 9156 | Cantor's theorem with the sethood requirement expressed as an antecedent. Theorem 23 of [Suppes] p. 97. (Contributed by NM, 7-Nov-2003.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ 𝒫 𝐴) | ||
Theorem | 2pwuninel 9157 | The power set of the power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. (Contributed by NM, 27-Jun-2008.) |
⊢ ¬ 𝒫 𝒫 ∪ 𝐴 ∈ 𝐴 | ||
Theorem | 2pwne 9158 | No set equals the power set of its power set. (Contributed by NM, 17-Nov-2008.) |
⊢ (𝐴 ∈ 𝑉 → 𝒫 𝒫 𝐴 ≠ 𝐴) | ||
Theorem | disjen 9159 | A stronger form of pwuninel 8281. We can use pwuninel 8281, 2pwuninel 9157 to create one or two sets disjoint from a given set 𝐴, but here we show that in fact such constructions exist for arbitrarily large disjoint extensions, which is to say that for any set 𝐵 we can construct a set 𝑥 that is equinumerous to it and disjoint from 𝐴. (Contributed by Mario Carneiro, 7-Feb-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ∪ ran 𝐴}) ≈ 𝐵)) | ||
Theorem | disjenex 9160* | Existence version of disjen 9159. (Contributed by Mario Carneiro, 7-Feb-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑥((𝐴 ∩ 𝑥) = ∅ ∧ 𝑥 ≈ 𝐵)) | ||
Theorem | domss2 9161 | A corollary of disjenex 9160. If 𝐹 is an injection from 𝐴 to 𝐵 then 𝐺 is a right inverse of 𝐹 from 𝐵 to a superset of 𝐴. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.) |
⊢ 𝐺 = ◡(𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) ⇒ ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐺:𝐵–1-1-onto→ran 𝐺 ∧ 𝐴 ⊆ ran 𝐺 ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))) | ||
Theorem | domssex2 9162* | A corollary of disjenex 9160. If 𝐹 is an injection from 𝐴 to 𝐵 then there is a right inverse 𝑔 of 𝐹 from 𝐵 to a superset of 𝐴. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.) |
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑔(𝑔:𝐵–1-1→V ∧ (𝑔 ∘ 𝐹) = ( I ↾ 𝐴))) | ||
Theorem | domssex 9163* | Weakening of domssex2 9162 to forget the functions in favor of dominance and equinumerosity. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.) |
⊢ (𝐴 ≼ 𝐵 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝐵 ≈ 𝑥)) | ||
Theorem | xpf1o 9164* | Construct a bijection on a Cartesian product given bijections on the factors. (Contributed by Mario Carneiro, 30-May-2015.) |
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑋):𝐴–1-1-onto→𝐵) & ⊢ (𝜑 → (𝑦 ∈ 𝐶 ↦ 𝑌):𝐶–1-1-onto→𝐷) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 〈𝑋, 𝑌〉):(𝐴 × 𝐶)–1-1-onto→(𝐵 × 𝐷)) | ||
Theorem | xpen 9165 | Equinumerosity law for Cartesian product. Proposition 4.22(b) of [Mendelson] p. 254. (Contributed by NM, 24-Jul-2004.) (Proof shortened by Mario Carneiro, 26-Apr-2015.) |
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐶) ≈ (𝐵 × 𝐷)) | ||
Theorem | mapen 9166 | Two set exponentiations are equinumerous when their bases and exponents are equinumerous. Theorem 6H(c) of [Enderton] p. 139. (Contributed by NM, 16-Dec-2003.) (Proof shortened by Mario Carneiro, 26-Apr-2015.) |
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 ↑m 𝐶) ≈ (𝐵 ↑m 𝐷)) | ||
Theorem | mapdom1 9167 | Order-preserving property of set exponentiation. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 9-Mar-2013.) |
⊢ (𝐴 ≼ 𝐵 → (𝐴 ↑m 𝐶) ≼ (𝐵 ↑m 𝐶)) | ||
Theorem | mapxpen 9168 | Equinumerosity law for double set exponentiation. Proposition 10.45 of [TakeutiZaring] p. 96. (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 24-Jun-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 ↑m 𝐵) ↑m 𝐶) ≈ (𝐴 ↑m (𝐵 × 𝐶))) | ||
Theorem | xpmapenlem 9169* | Lemma for xpmapen 9170. (Contributed by NM, 1-May-2004.) (Revised by Mario Carneiro, 16-Nov-2014.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 = (𝑧 ∈ 𝐶 ↦ (1st ‘(𝑥‘𝑧))) & ⊢ 𝑅 = (𝑧 ∈ 𝐶 ↦ (2nd ‘(𝑥‘𝑧))) & ⊢ 𝑆 = (𝑧 ∈ 𝐶 ↦ 〈((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)〉) ⇒ ⊢ ((𝐴 × 𝐵) ↑m 𝐶) ≈ ((𝐴 ↑m 𝐶) × (𝐵 ↑m 𝐶)) | ||
Theorem | xpmapen 9170 | Equinumerosity law for set exponentiation of a Cartesian product. Exercise 4.47 of [Mendelson] p. 255. (Contributed by NM, 23-Feb-2004.) (Proof shortened by Mario Carneiro, 16-Nov-2014.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ ((𝐴 × 𝐵) ↑m 𝐶) ≈ ((𝐴 ↑m 𝐶) × (𝐵 ↑m 𝐶)) | ||
Theorem | mapunen 9171 | Equinumerosity law for set exponentiation of a disjoint union. Exercise 4.45 of [Mendelson] p. 255. (Contributed by NM, 23-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐶 ↑m (𝐴 ∪ 𝐵)) ≈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵))) | ||
Theorem | map2xp 9172 | A cardinal power with exponent 2 is equivalent to a Cartesian product with itself. (Contributed by Mario Carneiro, 31-May-2015.) (Proof shortened by AV, 17-Jul-2022.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m 2o) ≈ (𝐴 × 𝐴)) | ||
Theorem | mapdom2 9173 | Order-preserving property of set exponentiation. Theorem 6L(d) of [Enderton] p. 149. (Contributed by NM, 23-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ ((𝐴 ≼ 𝐵 ∧ ¬ (𝐴 = ∅ ∧ 𝐶 = ∅)) → (𝐶 ↑m 𝐴) ≼ (𝐶 ↑m 𝐵)) | ||
Theorem | mapdom3 9174 | Set exponentiation dominates the base. (Contributed by Mario Carneiro, 30-Apr-2015.) (Proof shortened by AV, 17-Jul-2022.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴 ↑m 𝐵)) | ||
Theorem | pwen 9175 | If two sets are equinumerous, then their power sets are equinumerous. Proposition 10.15 of [TakeutiZaring] p. 87. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 9-Apr-2015.) |
⊢ (𝐴 ≈ 𝐵 → 𝒫 𝐴 ≈ 𝒫 𝐵) | ||
Theorem | ssenen 9176* | Equinumerosity of equinumerous subsets of a set. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 16-Nov-2014.) |
⊢ (𝐴 ≈ 𝐵 → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐶)} ≈ {𝑥 ∣ (𝑥 ⊆ 𝐵 ∧ 𝑥 ≈ 𝐶)}) | ||
Theorem | limenpsi 9177 | A limit ordinal is equinumerous to a proper subset of itself. (Contributed by NM, 30-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.) |
⊢ Lim 𝐴 ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ (𝐴 ∖ {∅})) | ||
Theorem | limensuci 9178 | A limit ordinal is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.) |
⊢ Lim 𝐴 ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ suc 𝐴) | ||
Theorem | limensuc 9179 | A limit ordinal is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.) |
⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → 𝐴 ≈ suc 𝐴) | ||
Theorem | infensuc 9180 | Any infinite ordinal is equinumerous to its successor. Exercise 7 of [TakeutiZaring] p. 88. Proved without the Axiom of Infinity. (Contributed by NM, 30-Oct-2003.) (Revised by Mario Carneiro, 13-Jan-2013.) |
⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → 𝐴 ≈ suc 𝐴) | ||
Theorem | dif1enlem 9181 | Lemma for rexdif1en 9183 and dif1en 9185. (Contributed by BTernaryTau, 18-Aug-2024.) Generalize to all ordinals and add a sethood requirement to avoid ax-un 7741. (Revised by BTernaryTau, 5-Jan-2025.) |
⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝑀 ∈ On) ∧ 𝐹:𝐴–1-1-onto→suc 𝑀) → (𝐴 ∖ {(◡𝐹‘𝑀)}) ≈ 𝑀) | ||
Theorem | dif1enlemOLD 9182 | Obsolete version of dif1enlem 9181 as of 5-Jan-2025. (Contributed by BTernaryTau, 18-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝐹 ∈ 𝑉 ∧ 𝑀 ∈ ω ∧ 𝐹:𝐴–1-1-onto→suc 𝑀) → (𝐴 ∖ {(◡𝐹‘𝑀)}) ≈ 𝑀) | ||
Theorem | rexdif1en 9183* | If a set is equinumerous to a nonzero ordinal, then there exists an element in that set such that removing it leaves the set equinumerous to the predecessor of that ordinal. (Contributed by BTernaryTau, 26-Aug-2024.) Generalize to all ordinals and avoid ax-un 7741. (Revised by BTernaryTau, 5-Jan-2025.) |
⊢ ((𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀) | ||
Theorem | rexdif1enOLD 9184* | Obsolete version of rexdif1en 9183 as of 5-Jan-2025. (Contributed by BTernaryTau, 26-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀) | ||
Theorem | dif1en 9185 | If a set 𝐴 is equinumerous to the successor of an ordinal 𝑀, then 𝐴 with an element removed is equinumerous to 𝑀. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.) Avoid ax-pow 5365. (Revised by BTernaryTau, 26-Aug-2024.) Generalize to all ordinals. (Revised by BTernaryTau, 6-Jan-2025.) |
⊢ ((𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀) | ||
Theorem | dif1ennn 9186 | If a set 𝐴 is equinumerous to the successor of a natural number 𝑀, then 𝐴 with an element removed is equinumerous to 𝑀. See also dif1ennnALT 9302. (Contributed by BTernaryTau, 6-Jan-2025.) |
⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀) | ||
Theorem | dif1enOLD 9187 | Obsolete version of dif1en 9185 as of 6-Jan-2025. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.) Avoid ax-pow 5365. (Revised by BTernaryTau, 26-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀) | ||
Theorem | findcard 9188* | Schema for induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on the given set with any one element removed. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = (𝑦 ∖ {𝑧}) → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) & ⊢ 𝜓 & ⊢ (𝑦 ∈ Fin → (∀𝑧 ∈ 𝑦 𝜒 → 𝜃)) ⇒ ⊢ (𝐴 ∈ Fin → 𝜏) | ||
Theorem | findcard2 9189* | Schema for induction on the cardinality of a finite set. The inductive step shows that the result is true if one more element is added to the set. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 8-Jul-2010.) Avoid ax-pow 5365. (Revised by BTernaryTau, 26-Aug-2024.) |
⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) & ⊢ 𝜓 & ⊢ (𝑦 ∈ Fin → (𝜒 → 𝜃)) ⇒ ⊢ (𝐴 ∈ Fin → 𝜏) | ||
Theorem | findcard2s 9190* | Variation of findcard2 9189 requiring that the element added in the induction step not be a member of the original set. (Contributed by Paul Chapman, 30-Nov-2012.) |
⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) & ⊢ 𝜓 & ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (𝜒 → 𝜃)) ⇒ ⊢ (𝐴 ∈ Fin → 𝜏) | ||
Theorem | findcard2d 9191* | Deduction version of findcard2 9189. (Contributed by SO, 16-Jul-2018.) |
⊢ (𝑥 = ∅ → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) & ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜓 ↔ 𝜏)) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂)) & ⊢ (𝜑 → 𝜒) & ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝜃 → 𝜏)) & ⊢ (𝜑 → 𝐴 ∈ Fin) ⇒ ⊢ (𝜑 → 𝜂) | ||
Theorem | nnfi 9192 | Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015.) Avoid ax-pow 5365. (Revised by BTernaryTau, 23-Sep-2024.) |
⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin) | ||
Theorem | pssnn 9193* | A proper subset of a natural number is equinumerous to some smaller number. Lemma 6F of [Enderton] p. 137. (Contributed by NM, 22-Jun-1998.) (Revised by Mario Carneiro, 16-Nov-2014.) Avoid ax-pow 5365. (Revised by BTernaryTau, 31-Jul-2024.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥) | ||
Theorem | ssnnfi 9194 | A subset of a natural number is finite. (Contributed by NM, 24-Jun-1998.) (Proof shortened by BTernaryTau, 23-Sep-2024.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin) | ||
Theorem | ssnnfiOLD 9195 | Obsolete version of ssnnfi 9194 as of 23-Sep-2024. (Contributed by NM, 24-Jun-1998.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin) | ||
Theorem | 0fin 9196 | The empty set is finite. (Contributed by FL, 14-Jul-2008.) |
⊢ ∅ ∈ Fin | ||
Theorem | unfi 9197 | The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. (Contributed by NM, 16-Nov-2002.) Avoid ax-pow 5365. (Revised by BTernaryTau, 7-Aug-2024.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin) | ||
Theorem | ssfi 9198 | A subset of a finite set is finite. Corollary 6G of [Enderton] p. 138. For a shorter proof using ax-pow 5365, see ssfiALT 9199. (Contributed by NM, 24-Jun-1998.) Avoid ax-pow 5365. (Revised by BTernaryTau, 12-Aug-2024.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin) | ||
Theorem | ssfiALT 9199 | Shorter proof of ssfi 9198 using ax-pow 5365. (Contributed by NM, 24-Jun-1998.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin) | ||
Theorem | imafi 9200 | Images of finite sets are finite. For a shorter proof using ax-pow 5365, see imafiALT 9371. (Contributed by Stefan O'Rear, 22-Feb-2015.) Avoid ax-pow 5365. (Revised by BTernaryTau, 7-Sep-2024.) |
⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → (𝐹 “ 𝑋) ∈ Fin) |
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