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Theorem en3i 8036
 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 8034 . 2 (⊤ → 𝐴𝐵)
1211trud 1533 1 𝐴𝐵
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523  ⊤wtru 1524   ∈ wcel 2030  Vcvv 3231   class class class wbr 4685   ≈ cen 7994 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-en 7998 This theorem is referenced by:  xpmapenlem  8168  nn0ennn  12818
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