Theorem disjex 5823

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 5744	. . 3 ⊢ Disj = {⟨x, y⟩ ∣ (x ∩ y) = ∅}
2		oteltxp 5782	. . . . . . . . . 10 ⊢ (⟨{z}, ⟨x, y⟩⟩ ∈ ( S ⊗ S ) ↔ (⟨{z}, x⟩ ∈ S ∧ ⟨{z}, y⟩ ∈ S ))
3		vex 2862	. . . . . . . . . . . 12 ⊢ z ∈ V
4		vex 2862	. . . . . . . . . . . 12 ⊢ x ∈ V
5	3, 4	opelssetsn 4760	. . . . . . . . . . 11 ⊢ (⟨{z}, x⟩ ∈ S ↔ z ∈ x)
6		vex 2862	. . . . . . . . . . . 12 ⊢ y ∈ V
7	3, 6	opelssetsn 4760	. . . . . . . . . . 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 4947	. . . . . . . 8 ⊢ (⟨x, y⟩ ∈ (( S ⊗ S ) “ 1c) ↔ ∃z⟨{z}, ⟨x, y⟩⟩ ∈ ( S ⊗ S ))
12		df-rex 2620	. . . . . . . 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 2627	. . . . . . 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 3591	. . . . 5 ⊢ ((x ∩ y) = ∅ ↔ ∀z ∈ x ¬ z ∈ y)
18	4, 6	opex 4588	. . . . . 6 ⊢ ⟨x, y⟩ ∈ V
19	18	elcompl 3225	. . . . 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 4866	. . 3 ⊢ ∼ (( S ⊗ S ) “ 1c) = {⟨x, y⟩ ∣ (x ∩ y) = ∅}
22	1, 21	eqtr4i 2376	. 2 ⊢ Disj = ∼ (( S ⊗ S ) “ 1c)
23		ssetex 4744	. . . . 5 ⊢ S ∈ V
24	23, 23	txpex 5785	. . . 4 ⊢ ( S ⊗ S ) ∈ V
25		1cex 4142	. . . 4 ⊢ 1c ∈ V
26	24, 25	imaex 4747	. . 3 ⊢ (( S ⊗ S ) “ 1c) ∈ V
27	26	complex 4104	. 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 2614  ∃wrex 2615  Vcvv 2859   ∼ ccompl 3205   ∩ cin 3208  ∅c0 3550  {csn 3737  1_cc1c 4134  ⟨cop 4561  {copab 4622   S csset 4719   “ cima 4722   ⊗ ctxp 5735   Disj cdisj 5743

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 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-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-cnv 4785  df-2nd 4797  df-txp 5736  df-disj 5744

This theorem is referenced by:  addcfnex  5824