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Mirrors > Home > MPE Home > Th. List > ensn1 | Structured version Visualization version GIF version |
Description: A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.) |
Ref | Expression |
---|---|
ensn1.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
ensn1 | ⊢ {𝐴} ≈ 1o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex 5297 | . . . 4 ⊢ {〈𝐴, ∅〉} ∈ V | |
2 | f1oeq1 6579 | . . . 4 ⊢ (𝑓 = {〈𝐴, ∅〉} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {〈𝐴, ∅〉}:{𝐴}–1-1-onto→{∅})) | |
3 | ensn1.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
4 | 0ex 5175 | . . . . 5 ⊢ ∅ ∈ V | |
5 | 3, 4 | f1osn 6629 | . . . 4 ⊢ {〈𝐴, ∅〉}:{𝐴}–1-1-onto→{∅} |
6 | 1, 2, 5 | ceqsexv2d 3490 | . . 3 ⊢ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅} |
7 | bren 8501 | . . 3 ⊢ ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}) | |
8 | 6, 7 | mpbir 234 | . 2 ⊢ {𝐴} ≈ {∅} |
9 | df1o2 8099 | . 2 ⊢ 1o = {∅} | |
10 | 8, 9 | breqtrri 5057 | 1 ⊢ {𝐴} ≈ 1o |
Colors of variables: wff setvar class |
Syntax hints: ∃wex 1781 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 {csn 4525 〈cop 4531 class class class wbr 5030 –1-1-onto→wf1o 6323 1oc1o 8078 ≈ cen 8489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-suc 6165 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-1o 8085 df-en 8493 |
This theorem is referenced by: ensn1g 8557 en1 8559 fodomfi 8781 pm54.43 9414 1nprm 16013 gex1 18708 sylow2a 18736 0frgp 18897 en1top 21589 en2top 21590 t1connperf 22041 ptcmplem2 22658 xrge0tsms2 23440 sconnpi1 32599 |
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