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	⊢ C ≠ D

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.

Step	Hyp	Ref	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 = C → ⟨y, z⟩ = ⟨y, C⟩)
5	4	eqeq2d 2364	. . . . . 6 ⊢ (z = C → (x = ⟨y, z⟩ ↔ x = ⟨y, C⟩))
6	3, 5	rexsn 3768	. . . . 5 ⊢ (∃z ∈ {C}x = ⟨y, z⟩ ↔ x = ⟨y, C⟩)
7	6	rexbii 2639	. . . 4 ⊢ (∃y ∈ A ∃z ∈ {C}x = ⟨y, z⟩ ↔ ∃y ∈ A x = ⟨y, C⟩)
8	2, 7	bitri 240	. . 3 ⊢ (x ∈ (A × {C}) ↔ ∃y ∈ A x = ⟨y, C⟩)
9		xpnedisj.2	. . . . . . . 8 ⊢ C ≠ D
10		df-ne 2518	. . . . . . . 8 ⊢ (C ≠ D ↔ ¬ C = D)
11	9, 10	mpbi 199	. . . . . . 7 ⊢ ¬ C = D
12		elsni 3757	. . . . . . 7 ⊢ (C ∈ {D} → C = D)
13	11, 12	mto 167	. . . . . 6 ⊢ ¬ C ∈ {D}
14	13	intnan 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}))
17	15, 16	syl6bb 252	. . . . 5 ⊢ (x = ⟨y, C⟩ → (x ∈ (B × {D}) ↔ (y ∈ B ∧ C ∈ {D})))
18	14, 17	mtbiri 294	. . . 4 ⊢ (x = ⟨y, C⟩ → ¬ x ∈ (B × {D}))
19	18	rexlimivw 2734	. . 3 ⊢ (∃y ∈ A x = ⟨y, C⟩ → ¬ x ∈ (B × {D}))
20	8, 19	sylbi 187	. 2 ⊢ (x ∈ (A × {C}) → ¬ x ∈ (B × {D}))
21	1, 20	mprgbir 2684	1 ⊢ ((A × {C}) ∩ (B × {D})) = ∅

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