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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 4752 | . 2 ⊢ Rel I | |
2 | vex 2740 | . . . . 5 ⊢ 𝑦 ∈ V | |
3 | 2 | ideq 4775 | . . . 4 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
4 | vex 2740 | . . . . 5 ⊢ 𝑥 ∈ V | |
5 | eqeng 6760 | . . . . 5 ⊢ (𝑥 ∈ V → (𝑥 = 𝑦 → 𝑥 ≈ 𝑦)) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝑥 ≈ 𝑦) |
7 | 3, 6 | sylbi 121 | . . 3 ⊢ (𝑥 I 𝑦 → 𝑥 ≈ 𝑦) |
8 | df-br 4001 | . . 3 ⊢ (𝑥 I 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ I ) | |
9 | df-br 4001 | . . 3 ⊢ (𝑥 ≈ 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ ≈ ) | |
10 | 7, 8, 9 | 3imtr3i 200 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ I → 〈𝑥, 𝑦〉 ∈ ≈ ) |
11 | 1, 10 | relssi 4714 | 1 ⊢ I ⊆ ≈ |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 Vcvv 2737 ⊆ wss 3129 〈cop 3594 class class class wbr 4000 I cid 4285 ≈ cen 6732 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-en 6735 |
This theorem is referenced by: (None) |
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