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Theorem en2i 8539
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 8537 . 2 (⊤ → 𝐴𝐵)
1211mptru 1545 1 𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wtru 1539  wcel 2115  Vcvv 3480   class class class wbr 5053  cen 8498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-en 8502
This theorem is referenced by:  xpsnen  8593  xpassen  8603
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