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Theorem en2i 7937
Description: Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 4-Jan-2004.)
Hypotheses
Ref Expression
en2i.1 𝐴 ∈ V
en2i.2 𝐵 ∈ V
en2i.3 (𝑥𝐴𝐶 ∈ V)
en2i.4 (𝑦𝐵𝐷 ∈ V)
en2i.5 ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷))
Assertion
Ref Expression
en2i 𝐴𝐵
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑦,𝐶   𝑥,𝐷
Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑦)

Proof of Theorem en2i
StepHypRef Expression
1 en2i.1 . . . 4 𝐴 ∈ V
21a1i 11 . . 3 (⊤ → 𝐴 ∈ V)
3 en2i.2 . . . 4 𝐵 ∈ V
43a1i 11 . . 3 (⊤ → 𝐵 ∈ V)
5 en2i.3 . . . 4 (𝑥𝐴𝐶 ∈ V)
65a1i 11 . . 3 (⊤ → (𝑥𝐴𝐶 ∈ V))
7 en2i.4 . . . 4 (𝑦𝐵𝐷 ∈ V)
87a1i 11 . . 3 (⊤ → (𝑦𝐵𝐷 ∈ V))
9 en2i.5 . . . 4 ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷))
109a1i 11 . . 3 (⊤ → ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷)))
112, 4, 6, 8, 10en2d 7935 . 2 (⊤ → 𝐴𝐵)
1211trud 1490 1 𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wtru 1481  wcel 1987  Vcvv 3186   class class class wbr 4613  cen 7896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-en 7900
This theorem is referenced by:  mapsnen  7979  xpsnen  7988  xpassen  7998
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