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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 4727 | . 2 ⊢ Rel I | |
2 | vex 2724 | . . . . 5 ⊢ 𝑦 ∈ V | |
3 | 2 | ideq 4750 | . . . 4 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
4 | vex 2724 | . . . . 5 ⊢ 𝑥 ∈ V | |
5 | eqeng 6723 | . . . . 5 ⊢ (𝑥 ∈ V → (𝑥 = 𝑦 → 𝑥 ≈ 𝑦)) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝑥 ≈ 𝑦) |
7 | 3, 6 | sylbi 120 | . . 3 ⊢ (𝑥 I 𝑦 → 𝑥 ≈ 𝑦) |
8 | df-br 3977 | . . 3 ⊢ (𝑥 I 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ I ) | |
9 | df-br 3977 | . . 3 ⊢ (𝑥 ≈ 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ ≈ ) | |
10 | 7, 8, 9 | 3imtr3i 199 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ I → 〈𝑥, 𝑦〉 ∈ ≈ ) |
11 | 1, 10 | relssi 4689 | 1 ⊢ I ⊆ ≈ |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2135 Vcvv 2721 ⊆ wss 3111 〈cop 3573 class class class wbr 3976 I cid 4260 ≈ cen 6695 |
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-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-en 6698 |
This theorem is referenced by: (None) |
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