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Theorem xpnedisj 5513
 Description: Cross products with non-equal singletons are disjoint. (Contributed by SF, 23-Feb-2015.)
Hypotheses
Ref Expression
xpnedisj.1 C V
xpnedisj.2 CD
Assertion
Ref Expression
xpnedisj ((A × {C}) ∩ (B × {D})) =

Proof of Theorem xpnedisj
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 disj 3591 . 2 (((A × {C}) ∩ (B × {D})) = x (A × {C}) ¬ x (B × {D}))
2 elxp2 4802 . . . 4 (x (A × {C}) ↔ y A z {C}x = y, z)
3 xpnedisj.1 . . . . . 6 C V
4 opeq2 4579 . . . . . . 7 (z = Cy, z = y, C)
54eqeq2d 2364 . . . . . 6 (z = C → (x = y, zx = y, C))
63, 5rexsn 3768 . . . . 5 (z {C}x = y, zx = y, C)
76rexbii 2639 . . . 4 (y A z {C}x = y, zy A x = y, C)
82, 7bitri 240 . . 3 (x (A × {C}) ↔ y A x = y, C)
9 xpnedisj.2 . . . . . . . 8 CD
10 df-ne 2518 . . . . . . . 8 (CD ↔ ¬ C = D)
119, 10mpbi 199 . . . . . . 7 ¬ C = D
12 elsni 3757 . . . . . . 7 (C {D} → C = D)
1311, 12mto 167 . . . . . 6 ¬ C {D}
1413intnan 880 . . . . 5 ¬ (y B C {D})
15 eleq1 2413 . . . . . 6 (x = y, C → (x (B × {D}) ↔ y, C (B × {D})))
16 opelxp 4811 . . . . . 6 (y, C (B × {D}) ↔ (y B C {D}))
1715, 16syl6bb 252 . . . . 5 (x = y, C → (x (B × {D}) ↔ (y B C {D})))
1814, 17mtbiri 294 . . . 4 (x = y, C → ¬ x (B × {D}))
1918rexlimivw 2734 . . 3 (y A x = y, C → ¬ x (B × {D}))
208, 19sylbi 187 . 2 (x (A × {C}) → ¬ x (B × {D}))
211, 20mprgbir 2684 1 ((A × {C}) ∩ (B × {D})) =
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∃wrex 2615  Vcvv 2859   ∩ cin 3208  ∅c0 3550  {csn 3737  ⟨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-xp 4784 This theorem is referenced by:  endisj  6051  ncaddccl  6144  tcdi  6164  ce0addcnnul  6179
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