Theorem disjex 5824

Description: The disjointedness relationship is a set. (Contributed by SF, 11-Feb-2015.)

Assertion

Ref	Expression
disjex	⊢ Disj ∈ V

Proof of Theorem disjex

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-disj 5745	. . 3 ⊢ Disj = {⟨x, y⟩ ∣ (x ∩ y) = ∅}
2		oteltxp 5783	. . . . . . . . . 10 ⊢ (⟨{z}, ⟨x, y⟩⟩ ∈ ( S ⊗ S ) ↔ (⟨{z}, x⟩ ∈ S ∧ ⟨{z}, y⟩ ∈ S ))
3		vex 2863	. . . . . . . . . . . 12 ⊢ z ∈ V
4		vex 2863	. . . . . . . . . . . 12 ⊢ x ∈ V
5	3, 4	opelssetsn 4761	. . . . . . . . . . 11 ⊢ (⟨{z}, x⟩ ∈ S ↔ z ∈ x)
6		vex 2863	. . . . . . . . . . . 12 ⊢ y ∈ V
7	3, 6	opelssetsn 4761	. . . . . . . . . . 11 ⊢ (⟨{z}, y⟩ ∈ S ↔ z ∈ y)
8	5, 7	anbi12i 678	. . . . . . . . . 10 ⊢ ((⟨{z}, x⟩ ∈ S ∧ ⟨{z}, y⟩ ∈ S ) ↔ (z ∈ x ∧ z ∈ y))
9	2, 8	bitri 240	. . . . . . . . 9 ⊢ (⟨{z}, ⟨x, y⟩⟩ ∈ ( S ⊗ S ) ↔ (z ∈ x ∧ z ∈ y))
10	9	exbii 1582	. . . . . . . 8 ⊢ (∃z⟨{z}, ⟨x, y⟩⟩ ∈ ( S ⊗ S ) ↔ ∃z(z ∈ x ∧ z ∈ y))
11		elima1c 4948	. . . . . . . 8 ⊢ (⟨x, y⟩ ∈ (( S ⊗ S ) “ 1c) ↔ ∃z⟨{z}, ⟨x, y⟩⟩ ∈ ( S ⊗ S ))
12		df-rex 2621	. . . . . . . 8 ⊢ (∃z ∈ x z ∈ y ↔ ∃z(z ∈ x ∧ z ∈ y))
13	10, 11, 12	3bitr4i 268	. . . . . . 7 ⊢ (⟨x, y⟩ ∈ (( S ⊗ S ) “ 1c) ↔ ∃z ∈ x z ∈ y)
14		dfrex2 2628	. . . . . . 7 ⊢ (∃z ∈ x z ∈ y ↔ ¬ ∀z ∈ x ¬ z ∈ y)
15	13, 14	bitri 240	. . . . . 6 ⊢ (⟨x, y⟩ ∈ (( S ⊗ S ) “ 1c) ↔ ¬ ∀z ∈ x ¬ z ∈ y)
16	15	con2bii 322	. . . . 5 ⊢ (∀z ∈ x ¬ z ∈ y ↔ ¬ ⟨x, y⟩ ∈ (( S ⊗ S ) “ 1c))
17		disj 3592	. . . . 5 ⊢ ((x ∩ y) = ∅ ↔ ∀z ∈ x ¬ z ∈ y)
18	4, 6	opex 4589	. . . . . 6 ⊢ ⟨x, y⟩ ∈ V
19	18	elcompl 3226	. . . . 5 ⊢ (⟨x, y⟩ ∈ ∼ (( S ⊗ S ) “ 1c) ↔ ¬ ⟨x, y⟩ ∈ (( S ⊗ S ) “ 1c))
20	16, 17, 19	3bitr4ri 269	. . . 4 ⊢ (⟨x, y⟩ ∈ ∼ (( S ⊗ S ) “ 1c) ↔ (x ∩ y) = ∅)
21	20	opabbi2i 4867	. . 3 ⊢ ∼ (( S ⊗ S ) “ 1c) = {⟨x, y⟩ ∣ (x ∩ y) = ∅}
22	1, 21	eqtr4i 2376	. 2 ⊢ Disj = ∼ (( S ⊗ S ) “ 1c)
23		ssetex 4745	. . . . 5 ⊢ S ∈ V
24	23, 23	txpex 5786	. . . 4 ⊢ ( S ⊗ S ) ∈ V
25		1cex 4143	. . . 4 ⊢ 1c ∈ V
26	24, 25	imaex 4748	. . 3 ⊢ (( S ⊗ S ) “ 1c) ∈ V
27	26	complex 4105	. 2 ⊢ ∼ (( S ⊗ S ) “ 1c) ∈ V
28	22, 27	eqeltri 2423	1 ⊢ Disj ∈ V

Syntax hints:  ¬ wn 3   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616  Vcvv 2860   ∼ ccompl 3206   ∩ cin 3209  ∅c0 3551  {csn 3738  1_cc1c 4135  ⟨cop 4562  {copab 4623   S csset 4720   “ cima 4723   ⊗ ctxp 5736   Disj cdisj 5744

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-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-cnv 4786  df-2nd 4798  df-txp 5737  df-disj 5745

This theorem is referenced by:  addcfnex  5825