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Mirrors > Home > MPE Home > Th. List > idssen | Structured version Visualization version 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 5701 | . 2 ⊢ Rel I | |
2 | vex 3500 | . . . . 5 ⊢ 𝑦 ∈ V | |
3 | 2 | ideq 5726 | . . . 4 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
4 | eqeng 8546 | . . . . 5 ⊢ (𝑥 ∈ V → (𝑥 = 𝑦 → 𝑥 ≈ 𝑦)) | |
5 | 4 | elv 3502 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝑥 ≈ 𝑦) |
6 | 3, 5 | sylbi 219 | . . 3 ⊢ (𝑥 I 𝑦 → 𝑥 ≈ 𝑦) |
7 | df-br 5070 | . . 3 ⊢ (𝑥 I 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ I ) | |
8 | df-br 5070 | . . 3 ⊢ (𝑥 ≈ 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ ≈ ) | |
9 | 6, 7, 8 | 3imtr3i 293 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ I → 〈𝑥, 𝑦〉 ∈ ≈ ) |
10 | 1, 9 | relssi 5663 | 1 ⊢ I ⊆ ≈ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 Vcvv 3497 ⊆ wss 3939 〈cop 4576 class class class wbr 5069 I cid 5462 ≈ cen 8509 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-en 8513 |
This theorem is referenced by: (None) |
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