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Theorem xpm 4688
Description: The cross product of inhabited classes is inhabited. (Contributed by Jim Kingdon, 13-Dec-2018.)
Assertion
Ref Expression
xpm ((x x A y y B) ↔ z z (A × B))
Distinct variable groups:   x,A   y,B   z,A   z,B
Allowed substitution hints:   A(y)   B(x)

Proof of Theorem xpm
Dummy variables 𝑎 𝑏 w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpmlem 4687 . 2 ((𝑎 𝑎 A 𝑏 𝑏 B) ↔ w w (A × B))
2 eleq1 2097 . . . 4 (𝑎 = x → (𝑎 Ax A))
32cbvexv 1792 . . 3 (𝑎 𝑎 Ax x A)
4 eleq1 2097 . . . 4 (𝑏 = y → (𝑏 By B))
54cbvexv 1792 . . 3 (𝑏 𝑏 By y B)
63, 5anbi12i 433 . 2 ((𝑎 𝑎 A 𝑏 𝑏 B) ↔ (x x A y y B))
7 eleq1 2097 . . 3 (w = z → (w (A × B) ↔ z (A × B)))
87cbvexv 1792 . 2 (w w (A × B) ↔ z z (A × B))
91, 6, 83bitr3i 199 1 ((x x A y y B) ↔ z z (A × B))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98  wex 1378   wcel 1390   × cxp 4286
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-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
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  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-opab 3810  df-xp 4294
This theorem is referenced by:  ssxpbm  4699  xp11m  4702  xpexr2m  4705  unixpm  4796
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