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Mirrors > Home > MPE Home > Th. List > ensn1g | Structured version Visualization version GIF version |
Description: A singleton is equinumerous to ordinal one. (Contributed by NM, 23-Apr-2004.) |
Ref | Expression |
---|---|
ensn1g | ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 4634 | . . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
2 | 1 | breq1d 5154 | . 2 ⊢ (𝑥 = 𝐴 → ({𝑥} ≈ 1o ↔ {𝐴} ≈ 1o)) |
3 | vex 3479 | . . 3 ⊢ 𝑥 ∈ V | |
4 | 3 | ensn1 9005 | . 2 ⊢ {𝑥} ≈ 1o |
5 | 2, 4 | vtoclg 3555 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {csn 4624 class class class wbr 5144 1oc1o 8446 ≈ cen 8924 |
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 2704 ax-sep 5295 ax-nul 5302 ax-pr 5423 |
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 2535 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5145 df-opab 5207 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-suc 6362 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-1o 8453 df-en 8928 |
This theorem is referenced by: enpr1g 9008 en1b 9011 en1bOLD 9012 snmapen1 9027 en2snOLDOLD 9031 snfi 9032 enpr2dOLD 9038 snnen2oOLD 9215 sucxpdom 9243 en1eqsnOLD 9263 en1eqsnbi 9264 pr2nelemOLD 9985 prdom2 9988 dju1en 10153 triv1nsgd 19038 snct 31909 rngoueqz 36714 safesnsupfidom1o 42039 sn1dom 42148 |
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