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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 4641 | . . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
2 | 1 | breq1d 5158 | . 2 ⊢ (𝑥 = 𝐴 → ({𝑥} ≈ 1o ↔ {𝐴} ≈ 1o)) |
3 | vex 3482 | . . 3 ⊢ 𝑥 ∈ V | |
4 | 3 | ensn1 9060 | . 2 ⊢ {𝑥} ≈ 1o |
5 | 2, 4 | vtoclg 3554 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {csn 4631 class class class wbr 5148 1oc1o 8498 ≈ cen 8981 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-suc 6392 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-1o 8505 df-en 8985 |
This theorem is referenced by: enpr1g 9062 en1b 9064 snmapen1 9078 snfi 9082 snfiOLD 9083 enpr2dOLD 9089 snnen2oOLD 9262 sucxpdom 9289 en1eqsnOLD 9307 en1eqsnbi 9308 pr2nelemOLD 10041 prdom2 10044 dju1en 10210 triv1nsgd 19204 snct 32731 rngoueqz 37927 safesnsupfidom1o 43407 sn1dom 43516 |
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