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Theorem xpnz 5045
 Description: The cross product of nonempty classes is nonempty. (Variation of a theorem contributed by Raph Levien, 30-Jun-2006.) (Contributed by set.mm contributors, 30-Jun-2006.) (Revised by set.mm contributors, 19-Apr-2007.)
Assertion
Ref Expression
xpnz ((A B) ↔ (A × B) ≠ )

Proof of Theorem xpnz
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0 3559 . . . . 5 (Ax x A)
2 n0 3559 . . . . 5 (By y B)
31, 2anbi12i 678 . . . 4 ((A B) ↔ (x x A y y B))
4 eeanv 1913 . . . 4 (xy(x A y B) ↔ (x x A y y B))
53, 4bitr4i 243 . . 3 ((A B) ↔ xy(x A y B))
6 opelxp 4811 . . . . 5 (x, y (A × B) ↔ (x A y B))
7 ne0i 3556 . . . . 5 (x, y (A × B) → (A × B) ≠ )
86, 7sylbir 204 . . . 4 ((x A y B) → (A × B) ≠ )
98exlimivv 1635 . . 3 (xy(x A y B) → (A × B) ≠ )
105, 9sylbi 187 . 2 ((A B) → (A × B) ≠ )
11 xpeq1 4798 . . . . 5 (A = → (A × B) = ( × B))
12 xp0r 4842 . . . . 5 ( × B) =
1311, 12syl6eq 2401 . . . 4 (A = → (A × B) = )
1413necon3i 2555 . . 3 ((A × B) ≠ A)
15 xpeq2 4799 . . . . 5 (B = → (A × B) = (A × ))
16 xp0 5044 . . . . 5 (A × ) =
1715, 16syl6eq 2401 . . . 4 (B = → (A × B) = )
1817necon3i 2555 . . 3 ((A × B) ≠ B)
1914, 18jca 518 . 2 ((A × B) ≠ → (A B))
2010, 19impbii 180 1 ((A B) ↔ (A × B) ≠ )
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∅c0 3550  ⟨cop 4561   × cxp 4770 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-xp 4784  df-cnv 4785 This theorem is referenced by:  xpeq0  5046  ssxpb  5055  xp11  5056  xpexr2  5110
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