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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 4523 | . 2 ⊢ Rel I | |
2 | vex 2615 | . . . . 5 ⊢ 𝑦 ∈ V | |
3 | 2 | ideq 4546 | . . . 4 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
4 | vex 2615 | . . . . 5 ⊢ 𝑥 ∈ V | |
5 | eqeng 6413 | . . . . 5 ⊢ (𝑥 ∈ V → (𝑥 = 𝑦 → 𝑥 ≈ 𝑦)) | |
6 | 4, 5 | ax-mp 7 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝑥 ≈ 𝑦) |
7 | 3, 6 | sylbi 119 | . . 3 ⊢ (𝑥 I 𝑦 → 𝑥 ≈ 𝑦) |
8 | df-br 3812 | . . 3 ⊢ (𝑥 I 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ I ) | |
9 | df-br 3812 | . . 3 ⊢ (𝑥 ≈ 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ ≈ ) | |
10 | 7, 8, 9 | 3imtr3i 198 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ I → 〈𝑥, 𝑦〉 ∈ ≈ ) |
11 | 1, 10 | relssi 4487 | 1 ⊢ I ⊆ ≈ |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1434 Vcvv 2612 ⊆ wss 2984 〈cop 3425 class class class wbr 3811 I cid 4079 ≈ cen 6385 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 4000 ax-un 4224 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-v 2614 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-br 3812 df-opab 3866 df-id 4084 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-rn 4412 df-res 4413 df-ima 4414 df-fun 4971 df-fn 4972 df-f 4973 df-f1 4974 df-fo 4975 df-f1o 4976 df-en 6388 |
This theorem is referenced by: (None) |
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