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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 8255 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ 𝐴) | |
2 | breq2 4878 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ≈ 𝐴 ↔ 𝐴 ≈ 𝐵)) | |
3 | 1, 2 | syl5ibcom 237 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → 𝐴 ≈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 class class class wbr 4874 ≈ cen 8220 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-en 8224 |
This theorem is referenced by: idssen 8268 nneneq 8413 onomeneq 8420 pr2ne 9142 alephord 9212 alephdom 9218 fin23lem25 9462 alephadd 9715 rp-isfinite5 38705 |
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