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Mirrors > Home > MPE Home > Th. List > en2sn | Structured version Visualization version GIF version |
Description: Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.) Avoid ax-pow 5371. (Revised by BTernaryTau, 31-Jul-2024.) Avoid ax-un 7754. (Revised by BTernaryTau, 25-Sep-2024.) |
Ref | Expression |
---|---|
en2sn | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴} ≈ {𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex 5442 | . . 3 ⊢ {〈𝐴, 𝐵〉} ∈ V | |
2 | f1osng 6890 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵}) | |
3 | f1oeq1 6837 | . . . 4 ⊢ (𝑓 = {〈𝐴, 𝐵〉} → (𝑓:{𝐴}–1-1-onto→{𝐵} ↔ {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵})) | |
4 | 3 | spcegv 3597 | . . 3 ⊢ ({〈𝐴, 𝐵〉} ∈ V → ({〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵} → ∃𝑓 𝑓:{𝐴}–1-1-onto→{𝐵})) |
5 | 1, 2, 4 | mpsyl 68 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ∃𝑓 𝑓:{𝐴}–1-1-onto→{𝐵}) |
6 | snex 5442 | . . 3 ⊢ {𝐴} ∈ V | |
7 | snex 5442 | . . 3 ⊢ {𝐵} ∈ V | |
8 | breng 8993 | . . 3 ⊢ (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} ≈ {𝐵} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{𝐵})) | |
9 | 6, 7, 8 | mp2an 692 | . 2 ⊢ ({𝐴} ≈ {𝐵} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{𝐵}) |
10 | 5, 9 | sylibr 234 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴} ≈ {𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∃wex 1776 ∈ wcel 2106 Vcvv 3478 {csn 4631 〈cop 4637 class class class wbr 5148 –1-1-onto→wf1o 6562 ≈ cen 8981 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-en 8985 |
This theorem is referenced by: enrefnn 9086 enpr2dOLD 9089 difsnen 9092 domunsncan 9111 sucdom2OLD 9121 domunsn 9166 limensuci 9192 infensuc 9194 unfi 9210 sucdom2 9241 0sdom1dom 9272 1sdom2dom 9281 dif1ennnALT 9309 fodomfi 9348 dif1card 10048 fin23lem26 10363 unsnen 10591 canthp1lem1 10690 fzennn 14006 hashsng 14405 mreexexlem4d 17692 |
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