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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 5365. (Revised by BTernaryTau, 31-Jul-2024.) Avoid ax-un 7755. (Revised by BTernaryTau, 25-Sep-2024.) |
| Ref | Expression |
|---|---|
| en2sn | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴} ≈ {𝐵}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | snex 5436 | . . 3 ⊢ {〈𝐴, 𝐵〉} ∈ V | |
| 2 | f1osng 6889 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵}) | |
| 3 | f1oeq1 6836 | . . . 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 5436 | . . 3 ⊢ {𝐴} ∈ V | |
| 7 | snex 5436 | . . 3 ⊢ {𝐵} ∈ V | |
| 8 | breng 8994 | . . 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 1779 ∈ wcel 2108 Vcvv 3480 {csn 4626 〈cop 4632 class class class wbr 5143 –1-1-onto→wf1o 6560 ≈ cen 8982 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-en 8986 |
| This theorem is referenced by: enrefnn 9087 enpr2dOLD 9090 difsnen 9093 domunsncan 9112 sucdom2OLD 9122 domunsn 9167 limensuci 9193 infensuc 9195 unfi 9211 sucdom2 9243 0sdom1dom 9274 1sdom2dom 9283 dif1ennnALT 9311 fodomfi 9350 dif1card 10050 fin23lem26 10365 unsnen 10593 canthp1lem1 10692 fzennn 14009 hashsng 14408 mreexexlem4d 17690 |
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