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| Mirrors > Home > MPE Home > Th. List > eqeng | Structured version Visualization version GIF version | ||
| Description: Equality implies equinumerosity. (Contributed by NM, 26-Oct-2003.) | 
| Ref | Expression | 
|---|---|
| eqeng | ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → 𝐴 ≈ 𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | enrefg 9024 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ 𝐴) | |
| 2 | breq2 5147 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ≈ 𝐴 ↔ 𝐴 ≈ 𝐵)) | |
| 3 | 1, 2 | syl5ibcom 245 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → 𝐴 ≈ 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ≈ cen 8982 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-en 8986 | 
| This theorem is referenced by: idssen 9037 nneneqOLD 9258 onomeneqOLD 9266 pr2neOLD 10045 alephord 10115 alephdom 10121 fin23lem25 10364 alephadd 10617 aks5lem7 42201 safesnsupfidom1o 43430 rp-isfinite5 43530 sn1dom 43539 prstchom2ALT 49168 | 
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