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Theorem en3i 8983
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 8981 . 2 (⊤ → 𝐴𝐵)
1211mptru 1540 1 𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wtru 1534  wcel 2098  Vcvv 3466   class class class wbr 5138  cen 8932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-en 8936
This theorem is referenced by:  xpmapenlem  9140  nn0ennn  13941
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