Theorem xpnedisj 5514

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 3592	. 2 ⊢ (((A × {C}) ∩ (B × {D})) = ∅ ↔ ∀x ∈ (A × {C}) ¬ x ∈ (B × {D}))
2		elxp2 4803	. . . 4 ⊢ (x ∈ (A × {C}) ↔ ∃y ∈ A ∃z ∈ {C}x = ⟨y, z⟩)
3		xpnedisj.1	. . . . . 6 ⊢ C ∈ V
4		opeq2 4580	. . . . . . 7 ⊢ (z = C → ⟨y, z⟩ = ⟨y, C⟩)
5	4	eqeq2d 2364	. . . . . 6 ⊢ (z = C → (x = ⟨y, z⟩ ↔ x = ⟨y, C⟩))
6	3, 5	rexsn 3769	. . . . 5 ⊢ (∃z ∈ {C}x = ⟨y, z⟩ ↔ x = ⟨y, C⟩)
7	6	rexbii 2640	. . . 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 2519	. . . . . . . 8 ⊢ (C ≠ D ↔ ¬ C = D)
11	9, 10	mpbi 199	. . . . . . 7 ⊢ ¬ C = D
12		elsni 3758	. . . . . . 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 4812	. . . . . 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 2735	. . 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 2685	1 ⊢ ((A × {C}) ∩ (B × {D})) = ∅

Syntax hints:  ¬ wn 3   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  ∃wrex 2616  Vcvv 2860   ∩ cin 3209  ∅c0 3551  {csn 3738  ⟨cop 4562   × cxp 4771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-xp 4785

This theorem is referenced by:  endisj  6052  ncaddccl  6145  tcdi  6165  ce0addcnnul  6180