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Theorem en2i 6788
Description: Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 4-Jan-2004.)
Hypotheses
Ref Expression
en2i.1 𝐴 ∈ V
en2i.2 𝐵 ∈ V
en2i.3 (𝑥𝐴𝐶 ∈ V)
en2i.4 (𝑦𝐵𝐷 ∈ V)
en2i.5 ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷))
Assertion
Ref Expression
en2i 𝐴𝐵
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑦,𝐶   𝑥,𝐷
Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑦)

Proof of Theorem en2i
StepHypRef Expression
1 en2i.1 . . . 4 𝐴 ∈ V
21a1i 9 . . 3 (⊤ → 𝐴 ∈ V)
3 en2i.2 . . . 4 𝐵 ∈ V
43a1i 9 . . 3 (⊤ → 𝐵 ∈ V)
5 en2i.3 . . . 4 (𝑥𝐴𝐶 ∈ V)
65a1i 9 . . 3 (⊤ → (𝑥𝐴𝐶 ∈ V))
7 en2i.4 . . . 4 (𝑦𝐵𝐷 ∈ V)
87a1i 9 . . 3 (⊤ → (𝑦𝐵𝐷 ∈ V))
9 en2i.5 . . . 4 ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷))
109a1i 9 . . 3 (⊤ → ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷)))
112, 4, 6, 8, 10en2d 6786 . 2 (⊤ → 𝐴𝐵)
1211mptru 1373 1 𝐴𝐵
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wtru 1365  wcel 2160  Vcvv 2752   class class class wbr 4018  cen 6756
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-en 6759
This theorem is referenced by:  mapsnen  6829  xpsnen  6839  xpassen  6848
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