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Theorem en2i 6186
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 (x A𝐶 V)
en2i.4 (y B𝐷 V)
en2i.5 ((x A y = 𝐶) ↔ (y B x = 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 (x A𝐶 V)
65a1i 9 . . 3 ( ⊤ → (x A𝐶 V))
7 en2i.4 . . . 4 (y B𝐷 V)
87a1i 9 . . 3 ( ⊤ → (y B𝐷 V))
9 en2i.5 . . . 4 ((x A y = 𝐶) ↔ (y B x = 𝐷))
109a1i 9 . . 3 ( ⊤ → ((x A y = 𝐶) ↔ (y B x = 𝐷)))
112, 4, 6, 8, 10en2d 6184 . 2 ( ⊤ → AB)
1211trud 1251 1 AB
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wtru 1243   wcel 1390  Vcvv 2551   class class class wbr 3755  cen 6155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-en 6158
This theorem is referenced by:  xpsnen  6231  xpassen  6240
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