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Mirrors > Home > ILE Home > Th. List > idssen | GIF version |
Description: Equality implies equinumerosity. (Contributed by NM, 30-Apr-1998.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Ref | Expression |
---|---|
idssen | ⊢ I ⊆ ≈ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reli 4628 | . 2 ⊢ Rel I | |
2 | vex 2660 | . . . . 5 ⊢ 𝑦 ∈ V | |
3 | 2 | ideq 4651 | . . . 4 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
4 | vex 2660 | . . . . 5 ⊢ 𝑥 ∈ V | |
5 | eqeng 6614 | . . . . 5 ⊢ (𝑥 ∈ V → (𝑥 = 𝑦 → 𝑥 ≈ 𝑦)) | |
6 | 4, 5 | ax-mp 7 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝑥 ≈ 𝑦) |
7 | 3, 6 | sylbi 120 | . . 3 ⊢ (𝑥 I 𝑦 → 𝑥 ≈ 𝑦) |
8 | df-br 3896 | . . 3 ⊢ (𝑥 I 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ I ) | |
9 | df-br 3896 | . . 3 ⊢ (𝑥 ≈ 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ ≈ ) | |
10 | 7, 8, 9 | 3imtr3i 199 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ I → 〈𝑥, 𝑦〉 ∈ ≈ ) |
11 | 1, 10 | relssi 4590 | 1 ⊢ I ⊆ ≈ |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1463 Vcvv 2657 ⊆ wss 3037 〈cop 3496 class class class wbr 3895 I cid 4170 ≈ cen 6586 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rex 2396 df-v 2659 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-opab 3950 df-id 4175 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-res 4511 df-ima 4512 df-fun 5083 df-fn 5084 df-f 5085 df-f1 5086 df-fo 5087 df-f1o 5088 df-en 6589 |
This theorem is referenced by: (None) |
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