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Theorem en3i 6665
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 9 . . 3 (⊤ → 𝐴 ∈ V)
3 en3i.2 . . . 4 𝐵 ∈ V
43a1i 9 . . 3 (⊤ → 𝐵 ∈ V)
5 en3i.3 . . . 4 (𝑥𝐴𝐶𝐵)
65a1i 9 . . 3 (⊤ → (𝑥𝐴𝐶𝐵))
7 en3i.4 . . . 4 (𝑦𝐵𝐷𝐴)
87a1i 9 . . 3 (⊤ → (𝑦𝐵𝐷𝐴))
9 en3i.5 . . . 4 ((𝑥𝐴𝑦𝐵) → (𝑥 = 𝐷𝑦 = 𝐶))
109a1i 9 . . 3 (⊤ → ((𝑥𝐴𝑦𝐵) → (𝑥 = 𝐷𝑦 = 𝐶)))
112, 4, 6, 8, 10en3d 6663 . 2 (⊤ → 𝐴𝐵)
1211mptru 1340 1 𝐴𝐵
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wtru 1332  wcel 1480  Vcvv 2686   class class class wbr 3929  cen 6632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-en 6635
This theorem is referenced by:  xpmapenlem  6743  nn0ennn  10213  oddennn  11912  evenennn  11913  znnen  11918
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