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Mirrors > Home > ILE Home > Th. List > ensn1g | 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 3592 | . . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
2 | 1 | breq1d 3997 | . 2 ⊢ (𝑥 = 𝐴 → ({𝑥} ≈ 1o ↔ {𝐴} ≈ 1o)) |
3 | vex 2733 | . . 3 ⊢ 𝑥 ∈ V | |
4 | 3 | ensn1 6772 | . 2 ⊢ {𝑥} ≈ 1o |
5 | 2, 4 | vtoclg 2790 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1o) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 {csn 3581 class class class wbr 3987 1oc1o 6386 ≈ cen 6714 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-suc 4354 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-1o 6393 df-en 6717 |
This theorem is referenced by: enpr1g 6774 en1bg 6776 en2sn 6789 snfig 6790 enpr2d 6793 snnen2og 6835 en1eqsn 6923 en1eqsnbi 6924 pr2nelem 7161 dju1en 7183 |
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