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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 4636 | . . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
| 2 | 1 | breq1d 5153 | . 2 ⊢ (𝑥 = 𝐴 → ({𝑥} ≈ 1o ↔ {𝐴} ≈ 1o)) |
| 3 | vex 3484 | . . 3 ⊢ 𝑥 ∈ V | |
| 4 | 3 | ensn1 9061 | . 2 ⊢ {𝑥} ≈ 1o |
| 5 | 2, 4 | vtoclg 3554 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1o) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {csn 4626 class class class wbr 5143 1oc1o 8499 ≈ cen 8982 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-suc 6390 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-1o 8506 df-en 8986 |
| This theorem is referenced by: enpr1g 9063 en1b 9065 snmapen1 9079 snfi 9083 snfiOLD 9084 enpr2dOLD 9090 snnen2oOLD 9264 sucxpdom 9291 en1eqsnOLD 9309 en1eqsnbi 9310 pr2nelemOLD 10043 prdom2 10046 dju1en 10212 triv1nsgd 19191 snct 32725 rngoueqz 37947 safesnsupfidom1o 43430 sn1dom 43539 |
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