![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5354. (Revised by BTernaryTau, 31-Jul-2024.) Avoid ax-un 7719. (Revised by BTernaryTau, 25-Sep-2024.) |
Ref | Expression |
---|---|
en2sn | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴} ≈ {𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex 5422 | . . 3 ⊢ {⟨𝐴, 𝐵⟩} ∈ V | |
2 | f1osng 6865 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}) | |
3 | f1oeq1 6812 | . . . 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 5422 | . . 3 ⊢ {𝐴} ∈ V | |
7 | snex 5422 | . . 3 ⊢ {𝐵} ∈ V | |
8 | breng 8945 | . . 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 4621 ⟨cop 4627 class class class wbr 5139 –1-1-onto→wf1o 6533 ≈ cen 8933 |
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 5290 ax-nul 5297 ax-pr 5418 |
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 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-br 5140 df-opab 5202 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-en 8937 |
This theorem is referenced by: enrefnn 9044 enpr2dOLD 9047 difsnen 9050 domunsncan 9069 sucdom2OLD 9079 domunsn 9124 limensuci 9150 infensuc 9152 unfi 9169 sucdom2 9203 0sdom1dom 9235 1sdom2dom 9244 dif1ennnALT 9274 dif1card 10002 fin23lem26 10317 unsnen 10545 canthp1lem1 10644 fzennn 13934 hashsng 14330 mreexexlem4d 17596 |
Copyright terms: Public domain | W3C validator |