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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 4535 | . . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
2 | 1 | breq1d 5040 | . 2 ⊢ (𝑥 = 𝐴 → ({𝑥} ≈ 1o ↔ {𝐴} ≈ 1o)) |
3 | vex 3444 | . . 3 ⊢ 𝑥 ∈ V | |
4 | 3 | ensn1 8556 | . 2 ⊢ {𝑥} ≈ 1o |
5 | 2, 4 | vtoclg 3515 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 {csn 4525 class class class wbr 5030 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: enpr1g 8558 en1b 8560 snmapen1 8574 en2sn 8576 snfi 8577 enpr2d 8580 snnen2o 8691 sucxpdom 8711 en1eqsn 8732 en1eqsnbi 8733 pr2nelem 9415 prdom2 9417 dju1en 9582 triv1nsgd 18317 snct 30475 rngoueqz 35378 sn1dom 40234 |
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