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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 3533 | . . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
2 | 1 | breq1d 3934 | . 2 ⊢ (𝑥 = 𝐴 → ({𝑥} ≈ 1o ↔ {𝐴} ≈ 1o)) |
3 | vex 2684 | . . 3 ⊢ 𝑥 ∈ V | |
4 | 3 | ensn1 6683 | . 2 ⊢ {𝑥} ≈ 1o |
5 | 2, 4 | vtoclg 2741 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1o) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 {csn 3522 class class class wbr 3924 1oc1o 6299 ≈ cen 6625 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-suc 4288 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-1o 6306 df-en 6628 |
This theorem is referenced by: enpr1g 6685 en1bg 6687 en2sn 6700 snfig 6701 enpr2d 6704 snnen2og 6746 en1eqsn 6829 en1eqsnbi 6830 pr2nelem 7040 dju1en 7062 |
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