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Theorem en3i 6542
 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 6540 . 2 (⊤ → 𝐴𝐵)
1211mptru 1299 1 𝐴𝐵
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1290  ⊤wtru 1291   ∈ wcel 1439  Vcvv 2620   class class class wbr 3851   ≈ cen 6509 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269 This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-en 6512 This theorem is referenced by:  xpmapenlem  6619  nn0ennn  9894  oddennn  11537  evenennn  11538  znnen  11543
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