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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 5325. (Revised by BTernaryTau, 31-Jul-2024.) Avoid ax-un 7677. (Revised by BTernaryTau, 25-Sep-2024.) |
Ref | Expression |
---|---|
en2sn | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴} ≈ {𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex 5393 | . . 3 ⊢ {⟨𝐴, 𝐵⟩} ∈ V | |
2 | f1osng 6830 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}) | |
3 | f1oeq1 6777 | . . . 4 ⊢ (𝑓 = {⟨𝐴, 𝐵⟩} → (𝑓:{𝐴}–1-1-onto→{𝐵} ↔ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵})) | |
4 | 3 | spcegv 3559 | . . 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 5393 | . . 3 ⊢ {𝐴} ∈ V | |
7 | snex 5393 | . . 3 ⊢ {𝐵} ∈ V | |
8 | breng 8899 | . . 3 ⊢ (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} ≈ {𝐵} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{𝐵})) | |
9 | 6, 7, 8 | mp2an 691 | . 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 397 ∃wex 1782 ∈ wcel 2107 Vcvv 3448 {csn 4591 ⟨cop 4597 class class class wbr 5110 –1-1-onto→wf1o 6500 ≈ cen 8887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-mo 2539 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-en 8891 |
This theorem is referenced by: enrefnn 8998 enpr2dOLD 9001 difsnen 9004 domunsncan 9023 sucdom2OLD 9033 domunsn 9078 limensuci 9104 infensuc 9106 unfi 9123 sucdom2 9157 0sdom1dom 9189 1sdom2dom 9198 dif1ennnALT 9228 dif1card 9953 fin23lem26 10268 unsnen 10496 canthp1lem1 10595 fzennn 13880 hashsng 14276 mreexexlem4d 17534 |
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