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Theorem en2i 8233
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 11 . . 3 (⊤ → 𝐴 ∈ V)
3 en2i.2 . . . 4 𝐵 ∈ V
43a1i 11 . . 3 (⊤ → 𝐵 ∈ V)
5 en2i.3 . . . 4 (𝑥𝐴𝐶 ∈ V)
65a1i 11 . . 3 (⊤ → (𝑥𝐴𝐶 ∈ V))
7 en2i.4 . . . 4 (𝑦𝐵𝐷 ∈ V)
87a1i 11 . . 3 (⊤ → (𝑦𝐵𝐷 ∈ V))
9 en2i.5 . . . 4 ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷))
109a1i 11 . . 3 (⊤ → ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷)))
112, 4, 6, 8, 10en2d 8231 . 2 (⊤ → 𝐴𝐵)
1211mptru 1661 1 𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wtru 1654  wcel 2157  Vcvv 3385   class class class wbr 4843  cen 8192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-en 8196
This theorem is referenced by:  xpsnen  8286  xpassen  8296
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