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Type | Label | Description |
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Statement | ||
Theorem | f1oenfi 9201 | If the domain of a one-to-one, onto function is finite, then the domain and range of the function are equinumerous. This theorem is proved without using the Axiom of Replacement or the Axiom of Power Sets (unlike f1oeng 8986). (Contributed by BTernaryTau, 8-Sep-2024.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) | ||
Theorem | f1oenfirn 9202 | If the range of a one-to-one, onto function is finite, then the domain and range of the function are equinumerous. (Contributed by BTernaryTau, 9-Sep-2024.) |
⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) | ||
Theorem | f1domfi 9203 | If the codomain of a one-to-one function is finite, then the function's domain is dominated by its codomain. This theorem is proved without using the Axiom of Replacement or the Axiom of Power Sets (unlike f1domg 8987). (Contributed by BTernaryTau, 25-Sep-2024.) |
⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵) | ||
Theorem | f1domfi2 9204 | If the domain of a one-to-one function is finite, then the function's domain is dominated by its codomain when the latter is a set. This theorem is proved without using the Axiom of Power Sets (unlike f1dom2g 8984). (Contributed by BTernaryTau, 24-Nov-2024.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵) | ||
Theorem | enreffi 9205 | Equinumerosity is reflexive for finite sets, proved without using the Axiom of Power Sets (unlike enrefg 8999). (Contributed by BTernaryTau, 8-Sep-2024.) |
⊢ (𝐴 ∈ Fin → 𝐴 ≈ 𝐴) | ||
Theorem | ensymfib 9206 | Symmetry of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike ensymb 9017). (Contributed by BTernaryTau, 9-Sep-2024.) |
⊢ (𝐴 ∈ Fin → (𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴)) | ||
Theorem | entrfil 9207 | Transitivity of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike entr 9021). (Contributed by BTernaryTau, 10-Sep-2024.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶) | ||
Theorem | enfii 9208 | A set equinumerous to a finite set is finite. (Contributed by Mario Carneiro, 12-Mar-2015.) Avoid ax-pow 5360. (Revised by BTernaryTau, 23-Sep-2024.) |
⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin) | ||
Theorem | enfi 9209 | Equinumerous sets have the same finiteness. For a shorter proof using ax-pow 5360, see enfiALT 9210. (Contributed by NM, 22-Aug-2008.) Avoid ax-pow 5360. (Revised by BTernaryTau, 23-Sep-2024.) |
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin)) | ||
Theorem | enfiALT 9210 | Shorter proof of enfi 9209 using ax-pow 5360. (Contributed by NM, 22-Aug-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin)) | ||
Theorem | domfi 9211 | A set dominated by a finite set is finite. (Contributed by NM, 23-Mar-2006.) (Revised by Mario Carneiro, 12-Mar-2015.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ Fin) | ||
Theorem | entrfi 9212 | Transitivity of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike entr 9021). (Contributed by BTernaryTau, 23-Sep-2024.) |
⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶) | ||
Theorem | entrfir 9213 | Transitivity of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike entr 9021). (Contributed by BTernaryTau, 23-Sep-2024.) |
⊢ ((𝐶 ∈ Fin ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶) | ||
Theorem | domtrfil 9214 | Transitivity of dominance relation when 𝐴 is finite, proved without using the Axiom of Power Sets (unlike domtr 9022). (Contributed by BTernaryTau, 24-Nov-2024.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) | ||
Theorem | domtrfi 9215 | Transitivity of dominance relation when 𝐵 is finite, proved without using the Axiom of Power Sets (unlike domtr 9022). (Contributed by BTernaryTau, 24-Nov-2024.) |
⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) | ||
Theorem | domtrfir 9216 | Transitivity of dominance relation for finite sets, proved without using the Axiom of Power Sets (unlike domtr 9022). (Contributed by BTernaryTau, 24-Nov-2024.) |
⊢ ((𝐶 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) | ||
Theorem | f1imaenfi 9217 | If a function is one-to-one, then the image of a finite subset of its domain under it is equinumerous to the subset. This theorem is proved without using the Axiom of Replacement or the Axiom of Power Sets (unlike f1imaeng 9029). (Contributed by BTernaryTau, 29-Sep-2024.) |
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ Fin) → (𝐹 “ 𝐶) ≈ 𝐶) | ||
Theorem | ssdomfi 9218 | A finite set dominates its subsets, proved without using the Axiom of Power Sets (unlike ssdomg 9015). (Contributed by BTernaryTau, 12-Nov-2024.) |
⊢ (𝐵 ∈ Fin → (𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵)) | ||
Theorem | ssdomfi2 9219 | A set dominates its finite subsets, proved without using the Axiom of Power Sets (unlike ssdomg 9015). (Contributed by BTernaryTau, 24-Nov-2024.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ≼ 𝐵) | ||
Theorem | sbthfilem 9220* | Lemma for sbthfi 9221. (Contributed by BTernaryTau, 4-Nov-2024.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} & ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) & ⊢ 𝐵 ∈ V ⇒ ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵) | ||
Theorem | sbthfi 9221 | Schroeder-Bernstein Theorem for finite sets, proved without using the Axiom of Power Sets (unlike sbth 9112). (Contributed by BTernaryTau, 4-Nov-2024.) |
⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵) | ||
Theorem | domnsymfi 9222 | If a set dominates a finite set, it cannot also be strictly dominated by the finite set. This theorem is proved without using the Axiom of Power Sets (unlike domnsym 9118). (Contributed by BTernaryTau, 22-Nov-2024.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≼ 𝐵) → ¬ 𝐵 ≺ 𝐴) | ||
Theorem | sdomdomtrfi 9223 | Transitivity of strict dominance and dominance when 𝐴 is finite, proved without using the Axiom of Power Sets (unlike sdomdomtr 9129). (Contributed by BTernaryTau, 25-Nov-2024.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≺ 𝐶) | ||
Theorem | domsdomtrfi 9224 | Transitivity of dominance and strict dominance when 𝐴 is finite, proved without using the Axiom of Power Sets (unlike domsdomtr 9131). (Contributed by BTernaryTau, 25-Nov-2024.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≺ 𝐶) | ||
Theorem | sucdom2 9225 | Strict dominance of a set over another set implies dominance over its successor. (Contributed by Mario Carneiro, 12-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.) Avoid ax-pow 5360. (Revised by BTernaryTau, 4-Dec-2024.) |
⊢ (𝐴 ≺ 𝐵 → suc 𝐴 ≼ 𝐵) | ||
Theorem | phplem1 9226 | Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor. (Contributed by NM, 26-May-1998.) Avoid ax-pow 5360. (Revised by BTernaryTau, 23-Sep-2024.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵})) | ||
Theorem | phplem2 9227 | Lemma for Pigeonhole Principle. Equinumerosity of successors implies equinumerosity of the original natural numbers. (Contributed by NM, 28-May-1998.) (Revised by Mario Carneiro, 24-Jun-2015.) Avoid ax-pow 5360. (Revised by BTernaryTau, 4-Nov-2024.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ≈ suc 𝐵 → 𝐴 ≈ 𝐵)) | ||
Theorem | nneneq 9228 | Two equinumerous natural numbers are equal. Proposition 10.20 of [TakeutiZaring] p. 90 and its converse. Also compare Corollary 6E of [Enderton] p. 136. (Contributed by NM, 28-May-1998.) Avoid ax-pow 5360. (Revised by BTernaryTau, 11-Nov-2024.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ≈ 𝐵 ↔ 𝐴 = 𝐵)) | ||
Theorem | php 9229 | Pigeonhole Principle. A natural number is not equinumerous to a proper subset of itself. Theorem (Pigeonhole Principle) of [Enderton] p. 134. The theorem is so-called because you can't put n + 1 pigeons into n holes (if each hole holds only one pigeon). The proof consists of phplem1 9226, phplem2 9227, nneneq 9228, and this final piece of the proof. (Contributed by NM, 29-May-1998.) Avoid ax-pow 5360. (Revised by BTernaryTau, 18-Nov-2024.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → ¬ 𝐴 ≈ 𝐵) | ||
Theorem | php2 9230 | Corollary of Pigeonhole Principle. (Contributed by NM, 31-May-1998.) Avoid ax-pow 5360. (Revised by BTernaryTau, 20-Nov-2024.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴) | ||
Theorem | php3 9231 | Corollary of Pigeonhole Principle. If 𝐴 is finite and 𝐵 is a proper subset of 𝐴, the 𝐵 is strictly less numerous than 𝐴. Stronger version of Corollary 6C of [Enderton] p. 135. (Contributed by NM, 22-Aug-2008.) Avoid ax-pow 5360. (Revised by BTernaryTau, 26-Nov-2024.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴) | ||
Theorem | php4 9232 | Corollary of the Pigeonhole Principle php 9229: a natural number is strictly dominated by its successor. (Contributed by NM, 26-Jul-2004.) |
⊢ (𝐴 ∈ ω → 𝐴 ≺ suc 𝐴) | ||
Theorem | php5 9233 | Corollary of the Pigeonhole Principle php 9229: a natural number is not equinumerous to its successor. Corollary 10.21(1) of [TakeutiZaring] p. 90. (Contributed by NM, 26-Jul-2004.) |
⊢ (𝐴 ∈ ω → ¬ 𝐴 ≈ suc 𝐴) | ||
Theorem | phpeqd 9234 | Corollary of the Pigeonhole Principle using equality. Strengthening of php 9229 expressed without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.) Avoid ax-pow. (Revised by BTernaryTau, 28-Nov-2024.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) & ⊢ (𝜑 → 𝐴 ≈ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | nndomog 9235 | Cardinal ordering agrees with ordinal number ordering when the smaller number is a natural number. Compare with nndomo 9252 when both are natural numbers. (Contributed by NM, 17-Jun-1998.) Generalize from nndomo 9252. (Revised by RP, 5-Nov-2023.) Avoid ax-pow 5360. (Revised by BTernaryTau, 29-Nov-2024.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ On) → (𝐴 ≼ 𝐵 ↔ 𝐴 ⊆ 𝐵)) | ||
Theorem | phplem1OLD 9236 | Obsolete lemma for php 9229 as of 22-Nov-2024. (Contributed by NM, 25-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ({𝐴} ∪ (𝐴 ∖ {𝐵})) = (suc 𝐴 ∖ {𝐵})) | ||
Theorem | phplem2OLD 9237 | Obsolete lemma for php 9229 as of 22-Nov-2024. (Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro, 16-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵})) | ||
Theorem | phplem3OLD 9238 | Obsolete version of phplem1 9226 as of 23-Sep-2024. (Contributed by NM, 26-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵})) | ||
Theorem | phplem4OLD 9239 | Obsolete version of phplem2 9227 as of 4-Nov-2024. (Contributed by NM, 28-May-1998.) (Revised by Mario Carneiro, 24-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ≈ suc 𝐵 → 𝐴 ≈ 𝐵)) | ||
Theorem | nneneqOLD 9240 | Obsolete version of nneneq 9228 as of 11-Nov-2024. (Contributed by NM, 28-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ≈ 𝐵 ↔ 𝐴 = 𝐵)) | ||
Theorem | phpOLD 9241 | Obsolete version of php 9229 as of 18-Nov-2024. (Contributed by NM, 29-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → ¬ 𝐴 ≈ 𝐵) | ||
Theorem | php2OLD 9242 | Obsolete version of php2 9230 as of 20-Nov-2024. (Contributed by NM, 31-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴) | ||
Theorem | php3OLD 9243 | Obsolete version of php3 9231 as of 26-Nov-2024. (Contributed by NM, 22-Aug-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴) | ||
Theorem | phpeqdOLD 9244 | Obsolete version of phpeqd 9234 as of 28-Nov-2024. (Contributed by Rohan Ridenour, 3-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) & ⊢ (𝜑 → 𝐴 ≈ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | nndomogOLD 9245 | Obsolete version of nndomog 9235 as of 29-Nov-2024. (Contributed by NM, 17-Jun-1998.) Generalize from nndomo 9252. (Revised by RP, 5-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ On) → (𝐴 ≼ 𝐵 ↔ 𝐴 ⊆ 𝐵)) | ||
Theorem | snnen2oOLD 9246 | Obsolete version of snnen2o 9256 as of 18-Nov-2024. (Contributed by AV, 6-Aug-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ¬ {𝐴} ≈ 2o | ||
Theorem | onomeneq 9247 | An ordinal number equinumerous to a natural number is equal to it. Proposition 10.22 of [TakeutiZaring] p. 90 and its converse. (Contributed by NM, 26-Jul-2004.) Avoid ax-pow 5360. (Revised by BTernaryTau, 2-Dec-2024.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ≈ 𝐵 ↔ 𝐴 = 𝐵)) | ||
Theorem | onomeneqOLD 9248 | Obsolete version of onomeneq 9247 as of 29-Nov-2024. (Contributed by NM, 26-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ≈ 𝐵 ↔ 𝐴 = 𝐵)) | ||
Theorem | onfin 9249 | An ordinal number is finite iff it is a natural number. Proposition 10.32 of [TakeutiZaring] p. 92. (Contributed by NM, 26-Jul-2004.) |
⊢ (𝐴 ∈ On → (𝐴 ∈ Fin ↔ 𝐴 ∈ ω)) | ||
Theorem | onfin2 9250 | A set is a natural number iff it is a finite ordinal. (Contributed by Mario Carneiro, 22-Jan-2013.) |
⊢ ω = (On ∩ Fin) | ||
Theorem | nnfiOLD 9251 | Obsolete version of nnfi 9186 as of 23-Sep-2024. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin) | ||
Theorem | nndomo 9252 | Cardinal ordering agrees with natural number ordering. Example 3 of [Enderton] p. 146. (Contributed by NM, 17-Jun-1998.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ≼ 𝐵 ↔ 𝐴 ⊆ 𝐵)) | ||
Theorem | nnsdomo 9253 | Cardinal ordering agrees with natural number ordering. (Contributed by NM, 17-Jun-1998.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ≺ 𝐵 ↔ 𝐴 ⊊ 𝐵)) | ||
Theorem | sucdom 9254 | Strict dominance of a set over a natural number is the same as dominance over its successor. (Contributed by Mario Carneiro, 12-Jan-2013.) Avoid ax-pow 5360. (Revised by BTernaryTau, 4-Dec-2024.) (Proof shortened by BJ, 11-Jan-2025.) |
⊢ (𝐴 ∈ ω → (𝐴 ≺ 𝐵 ↔ suc 𝐴 ≼ 𝐵)) | ||
Theorem | sucdomOLD 9255 | Obsolete version of sucdom 9254 as of 4-Dec-2024. (Contributed by Mario Carneiro, 12-Jan-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ω → (𝐴 ≺ 𝐵 ↔ suc 𝐴 ≼ 𝐵)) | ||
Theorem | snnen2o 9256 | A singleton {𝐴} is never equinumerous with the ordinal number 2. This holds for proper singletons (𝐴 ∈ V) as well as for singletons being the empty set (𝐴 ∉ V). (Contributed by AV, 6-Aug-2019.) Avoid ax-pow 5360, ax-un 7735. (Revised by BTernaryTau, 1-Dec-2024.) |
⊢ ¬ {𝐴} ≈ 2o | ||
Theorem | 0sdom1dom 9257 | Strict dominance over 0 is the same as dominance over 1. For a shorter proof requiring ax-un 7735, see 0sdom1domALT . (Contributed by NM, 28-Sep-2004.) Avoid ax-un 7735. (Revised by BTernaryTau, 7-Dec-2024.) |
⊢ (∅ ≺ 𝐴 ↔ 1o ≼ 𝐴) | ||
Theorem | 0sdom1domALT 9258 | Alternate proof of 0sdom1dom 9257, shorter but requiring ax-un 7735. (Contributed by NM, 28-Sep-2004.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∅ ≺ 𝐴 ↔ 1o ≼ 𝐴) | ||
Theorem | 1sdom2 9259 | Ordinal 1 is strictly dominated by ordinal 2. For a shorter proof requiring ax-un 7735, see 1sdom2ALT 9260. (Contributed by NM, 4-Apr-2007.) Avoid ax-un 7735. (Revised by BTernaryTau, 8-Dec-2024.) |
⊢ 1o ≺ 2o | ||
Theorem | 1sdom2ALT 9260 | Alternate proof of 1sdom2 9259, shorter but requiring ax-un 7735. (Contributed by NM, 4-Apr-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 1o ≺ 2o | ||
Theorem | sdom1 9261 | A set has less than one member iff it is empty. (Contributed by Stefan O'Rear, 28-Oct-2014.) Avoid ax-pow 5360, ax-un 7735. (Revised by BTernaryTau, 12-Dec-2024.) |
⊢ (𝐴 ≺ 1o ↔ 𝐴 = ∅) | ||
Theorem | sdom1OLD 9262 | Obsolete version of sdom1 9261 as of 12-Dec-2024. (Contributed by Stefan O'Rear, 28-Oct-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 ≺ 1o ↔ 𝐴 = ∅) | ||
Theorem | modom 9263 | Two ways to express "at most one". (Contributed by Stefan O'Rear, 28-Oct-2014.) |
⊢ (∃*𝑥𝜑 ↔ {𝑥 ∣ 𝜑} ≼ 1o) | ||
Theorem | modom2 9264* | Two ways to express "at most one". (Contributed by Mario Carneiro, 24-Dec-2016.) |
⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ 𝐴 ≼ 1o) | ||
Theorem | rex2dom 9265* | A set that has at least 2 different members dominates ordinal 2. (Contributed by BTernaryTau, 30-Dec-2024.) |
⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦) → 2o ≼ 𝐴) | ||
Theorem | 1sdom2dom 9266 | Strict dominance over 1 is the same as dominance over 2. (Contributed by BTernaryTau, 23-Dec-2024.) |
⊢ (1o ≺ 𝐴 ↔ 2o ≼ 𝐴) | ||
Theorem | 1sdom 9267* | A set that strictly dominates ordinal 1 has at least 2 different members. (Closely related to 2dom 9049.) (Contributed by Mario Carneiro, 12-Jan-2013.) Avoid ax-un 7735. (Revised by BTernaryTau, 30-Dec-2024.) |
⊢ (𝐴 ∈ 𝑉 → (1o ≺ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦)) | ||
Theorem | 1sdomOLD 9268* | Obsolete version of 1sdom 9267 as of 30-Dec-2024. (Contributed by Mario Carneiro, 12-Jan-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → (1o ≺ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦)) | ||
Theorem | unxpdomlem1 9269* | Lemma for unxpdom 9272. (Trivial substitution proof.) (Contributed by Mario Carneiro, 13-Jan-2013.) |
⊢ 𝐹 = (𝑥 ∈ (𝑎 ∪ 𝑏) ↦ 𝐺) & ⊢ 𝐺 = if(𝑥 ∈ 𝑎, 〈𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)〉, 〈if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥〉) ⇒ ⊢ (𝑧 ∈ (𝑎 ∪ 𝑏) → (𝐹‘𝑧) = if(𝑧 ∈ 𝑎, 〈𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)〉, 〈if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧〉)) | ||
Theorem | unxpdomlem2 9270* | Lemma for unxpdom 9272. (Contributed by Mario Carneiro, 13-Jan-2013.) |
⊢ 𝐹 = (𝑥 ∈ (𝑎 ∪ 𝑏) ↦ 𝐺) & ⊢ 𝐺 = if(𝑥 ∈ 𝑎, 〈𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)〉, 〈if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥〉) & ⊢ (𝜑 → 𝑤 ∈ (𝑎 ∪ 𝑏)) & ⊢ (𝜑 → ¬ 𝑚 = 𝑛) & ⊢ (𝜑 → ¬ 𝑠 = 𝑡) ⇒ ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) → ¬ (𝐹‘𝑧) = (𝐹‘𝑤)) | ||
Theorem | unxpdomlem3 9271* | Lemma for unxpdom 9272. (Contributed by Mario Carneiro, 13-Jan-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) |
⊢ 𝐹 = (𝑥 ∈ (𝑎 ∪ 𝑏) ↦ 𝐺) & ⊢ 𝐺 = if(𝑥 ∈ 𝑎, 〈𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)〉, 〈if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥〉) ⇒ ⊢ ((1o ≺ 𝑎 ∧ 1o ≺ 𝑏) → (𝑎 ∪ 𝑏) ≼ (𝑎 × 𝑏)) | ||
Theorem | unxpdom 9272 | Cartesian product dominates union for sets with cardinality greater than 1. Proposition 10.36 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 13-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.) |
⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐵)) | ||
Theorem | unxpdom2 9273 | Corollary of unxpdom 9272. (Contributed by NM, 16-Sep-2004.) |
⊢ ((1o ≺ 𝐴 ∧ 𝐵 ≼ 𝐴) → (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐴)) | ||
Theorem | sucxpdom 9274 | Cartesian product dominates successor for set with cardinality greater than 1. Proposition 10.38 of [TakeutiZaring] p. 93 (but generalized to arbitrary sets, not just ordinals). (Contributed by NM, 3-Sep-2004.) (Proof shortened by Mario Carneiro, 27-Apr-2015.) |
⊢ (1o ≺ 𝐴 → suc 𝐴 ≼ (𝐴 × 𝐴)) | ||
Theorem | pssinf 9275 | A set equinumerous to a proper subset of itself is infinite. Corollary 6D(a) of [Enderton] p. 136. (Contributed by NM, 2-Jun-1998.) |
⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐴 ≈ 𝐵) → ¬ 𝐵 ∈ Fin) | ||
Theorem | fisseneq 9276 | A finite set is equal to its subset if they are equinumerous. (Contributed by FL, 11-Aug-2008.) |
⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ 𝐴 ≈ 𝐵) → 𝐴 = 𝐵) | ||
Theorem | ominf 9277 | The set of natural numbers is infinite. Corollary 6D(b) of [Enderton] p. 136. (Contributed by NM, 2-Jun-1998.) Avoid ax-pow 5360. (Revised by BTernaryTau, 2-Jan-2025.) |
⊢ ¬ ω ∈ Fin | ||
Theorem | ominfOLD 9278 | Obsolete version of ominf 9277 as of 2-Jan-2025. (Contributed by NM, 2-Jun-1998.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ¬ ω ∈ Fin | ||
Theorem | isinf 9279* | Any set that is not finite is literally infinite, in the sense that it contains subsets of arbitrarily large finite cardinality. (It cannot be proven that the set has countably infinite subsets unless AC is invoked.) The proof does not require the Axiom of Infinity. (Contributed by Mario Carneiro, 15-Jan-2013.) Avoid ax-pow 5360. (Revised by BTernaryTau, 2-Jan-2025.) |
⊢ (¬ 𝐴 ∈ Fin → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛)) | ||
Theorem | isinfOLD 9280* | Obsolete version of isinf 9279 as of 2-Jan-2025. (Contributed by Mario Carneiro, 15-Jan-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ 𝐴 ∈ Fin → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛)) | ||
Theorem | fineqvlem 9281 | Lemma for fineqv 9282. (Contributed by Mario Carneiro, 20-Jan-2013.) (Proof shortened by Stefan O'Rear, 3-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ω ≼ 𝒫 𝒫 𝐴) | ||
Theorem | fineqv 9282 | If the Axiom of Infinity is denied, then all sets are finite (which implies the Axiom of Choice). (Contributed by Mario Carneiro, 20-Jan-2013.) (Revised by Mario Carneiro, 3-Jan-2015.) |
⊢ (¬ ω ∈ V ↔ Fin = V) | ||
Theorem | enfiiOLD 9283 | Obsolete version of enfii 9208 as of 23-Sep-2024. (Contributed by Mario Carneiro, 12-Mar-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin) | ||
Theorem | pssnnOLD 9284* | Obsolete version of pssnn 9187 as of 31-Jul-2024. (Contributed by NM, 22-Jun-1998.) (Revised by Mario Carneiro, 16-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥) | ||
Theorem | xpfir 9285 | The components of a nonempty finite Cartesian product are finite. (Contributed by Paul Chapman, 11-Apr-2009.) (Proof shortened by Mario Carneiro, 29-Apr-2015.) |
⊢ (((𝐴 × 𝐵) ∈ Fin ∧ (𝐴 × 𝐵) ≠ ∅) → (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin)) | ||
Theorem | ssfid 9286 | A subset of a finite set is finite, deduction version of ssfi 9192. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) ⇒ ⊢ (𝜑 → 𝐵 ∈ Fin) | ||
Theorem | infi 9287 | The intersection of two sets is finite if one of them is. (Contributed by Thierry Arnoux, 14-Feb-2017.) |
⊢ (𝐴 ∈ Fin → (𝐴 ∩ 𝐵) ∈ Fin) | ||
Theorem | rabfi 9288* | A restricted class built from a finite set is finite. (Contributed by Thierry Arnoux, 14-Feb-2017.) |
⊢ (𝐴 ∈ Fin → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ Fin) | ||
Theorem | finresfin 9289 | The restriction of a finite set is finite. (Contributed by Alexander van der Vekens, 3-Jan-2018.) |
⊢ (𝐸 ∈ Fin → (𝐸 ↾ 𝐵) ∈ Fin) | ||
Theorem | f1finf1o 9290 | Any injection from one finite set to another of equal size must be a bijection. (Contributed by Jeff Madsen, 5-Jun-2010.) (Revised by Mario Carneiro, 27-Feb-2014.) Avoid ax-pow 5360. (Revised by BTernaryTau, 4-Jan-2025.) |
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵)) | ||
Theorem | f1finf1oOLD 9291 | Obsolete version of f1finf1o 9290 as of 4-Jan-2025. (Contributed by Jeff Madsen, 5-Jun-2010.) (Revised by Mario Carneiro, 27-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵)) | ||
Theorem | nfielex 9292* | If a class is not finite, then it contains at least one element. (Contributed by Alexander van der Vekens, 12-Jan-2018.) |
⊢ (¬ 𝐴 ∈ Fin → ∃𝑥 𝑥 ∈ 𝐴) | ||
Theorem | en1eqsn 9293 | A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.) Avoid ax-pow 5360, ax-un 7735. (Revised by BTernaryTau, 4-Jan-2025.) |
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = {𝐴}) | ||
Theorem | en1eqsnOLD 9294 | Obsolete version of en1eqsn 9293 as of 4-Jan-2025. (Contributed by FL, 18-Aug-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = {𝐴}) | ||
Theorem | en1eqsnbi 9295 | A set containing an element has exactly one element iff it is a singleton. Formerly part of proof for rngen1zr 20659. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.) |
⊢ (𝐴 ∈ 𝐵 → (𝐵 ≈ 1o ↔ 𝐵 = {𝐴})) | ||
Theorem | dif1ennnALT 9296 | Alternate proof of dif1ennn 9180 using ax-pow 5360. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀) | ||
Theorem | enp1ilem 9297 | Lemma for uses of enp1i 9298. (Contributed by Mario Carneiro, 5-Jan-2016.) |
⊢ 𝑇 = ({𝑥} ∪ 𝑆) ⇒ ⊢ (𝑥 ∈ 𝐴 → ((𝐴 ∖ {𝑥}) = 𝑆 → 𝐴 = 𝑇)) | ||
Theorem | enp1i 9298* | Proof induction for en2 9300 and related theorems. (Contributed by Mario Carneiro, 5-Jan-2016.) Generalize to all ordinals and avoid ax-pow 5360, ax-un 7735. (Revised by BTernaryTau, 6-Jan-2025.) |
⊢ Ord 𝑀 & ⊢ 𝑁 = suc 𝑀 & ⊢ ((𝐴 ∖ {𝑥}) ≈ 𝑀 → 𝜑) & ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ⇒ ⊢ (𝐴 ≈ 𝑁 → ∃𝑥𝜓) | ||
Theorem | enp1iOLD 9299* | Obsolete version of enp1i 9298 as of 6-Jan-2025. (Contributed by Mario Carneiro, 5-Jan-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝑀 ∈ ω & ⊢ 𝑁 = suc 𝑀 & ⊢ ((𝐴 ∖ {𝑥}) ≈ 𝑀 → 𝜑) & ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ⇒ ⊢ (𝐴 ≈ 𝑁 → ∃𝑥𝜓) | ||
Theorem | en2 9300* | A set equinumerous to ordinal 2 is a pair. (Contributed by Mario Carneiro, 5-Jan-2016.) |
⊢ (𝐴 ≈ 2o → ∃𝑥∃𝑦 𝐴 = {𝑥, 𝑦}) |
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