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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 5353. (Revised by BTernaryTau, 31-Jul-2024.) Avoid ax-un 7718. (Revised by BTernaryTau, 25-Sep-2024.) |
Ref | Expression |
---|---|
en2sn | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴} ≈ {𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex 5421 | . . 3 ⊢ {〈𝐴, 𝐵〉} ∈ V | |
2 | f1osng 6864 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵}) | |
3 | f1oeq1 6811 | . . . 4 ⊢ (𝑓 = {〈𝐴, 𝐵〉} → (𝑓:{𝐴}–1-1-onto→{𝐵} ↔ {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵})) | |
4 | 3 | spcegv 3579 | . . 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 5421 | . . 3 ⊢ {𝐴} ∈ V | |
7 | snex 5421 | . . 3 ⊢ {𝐵} ∈ V | |
8 | breng 8943 | . . 3 ⊢ (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} ≈ {𝐵} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{𝐵})) | |
9 | 6, 7, 8 | mp2an 689 | . 2 ⊢ ({𝐴} ≈ {𝐵} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{𝐵}) |
10 | 5, 9 | sylibr 233 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴} ≈ {𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∃wex 1773 ∈ wcel 2098 Vcvv 3466 {csn 4620 〈cop 4626 class class class wbr 5138 –1-1-onto→wf1o 6532 ≈ cen 8931 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-mo 2526 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-br 5139 df-opab 5201 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-en 8935 |
This theorem is referenced by: enrefnn 9042 enpr2dOLD 9045 difsnen 9048 domunsncan 9067 sucdom2OLD 9077 domunsn 9122 limensuci 9148 infensuc 9150 unfi 9167 sucdom2 9201 0sdom1dom 9233 1sdom2dom 9242 dif1ennnALT 9272 dif1card 10000 fin23lem26 10315 unsnen 10543 canthp1lem1 10642 fzennn 13929 hashsng 14325 mreexexlem4d 17589 |
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