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Theorem en3i 8531
Description: Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 19-Jul-2004.)
Hypotheses
Ref Expression
en3i.1 𝐴 ∈ V
en3i.2 𝐵 ∈ V
en3i.3 (𝑥𝐴𝐶𝐵)
en3i.4 (𝑦𝐵𝐷𝐴)
en3i.5 ((𝑥𝐴𝑦𝐵) → (𝑥 = 𝐷𝑦 = 𝐶))
Assertion
Ref Expression
en3i 𝐴𝐵
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑦,𝐶   𝑥,𝐷
Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑦)

Proof of Theorem en3i
StepHypRef Expression
1 en3i.1 . . . 4 𝐴 ∈ V
21a1i 11 . . 3 (⊤ → 𝐴 ∈ V)
3 en3i.2 . . . 4 𝐵 ∈ V
43a1i 11 . . 3 (⊤ → 𝐵 ∈ V)
5 en3i.3 . . . 4 (𝑥𝐴𝐶𝐵)
65a1i 11 . . 3 (⊤ → (𝑥𝐴𝐶𝐵))
7 en3i.4 . . . 4 (𝑦𝐵𝐷𝐴)
87a1i 11 . . 3 (⊤ → (𝑦𝐵𝐷𝐴))
9 en3i.5 . . . 4 ((𝑥𝐴𝑦𝐵) → (𝑥 = 𝐷𝑦 = 𝐶))
109a1i 11 . . 3 (⊤ → ((𝑥𝐴𝑦𝐵) → (𝑥 = 𝐷𝑦 = 𝐶)))
112, 4, 6, 8, 10en3d 8529 . 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 2111  Vcvv 3441   class class class wbr 5030  cen 8489
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-en 8493
This theorem is referenced by:  xpmapenlem  8668  nn0ennn  13342
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