Theorem refex 5912

Description: The class of all reflexive relationships is a set. (Contributed by SF, 11-Mar-2015.)

Assertion

Ref	Expression
refex	⊢ Ref ∈ V

Proof of Theorem refex

Dummy variables p a r x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ref 5901	. . 3 ⊢ Ref = {⟨r, a⟩ ∣ ∀x ∈ a xrx}
2		vex 2863	. . . . . . 7 ⊢ r ∈ V
3		vex 2863	. . . . . . 7 ⊢ a ∈ V
4	2, 3	opex 4589	. . . . . 6 ⊢ ⟨r, a⟩ ∈ V
5	4	elcompl 3226	. . . . 5 ⊢ (⟨r, a⟩ ∈ ∼ (( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ) “ 1c) ↔ ¬ ⟨r, a⟩ ∈ (( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ) “ 1c))
6		elima1c 4948	. . . . . . . 8 ⊢ (⟨r, a⟩ ∈ (( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ) “ 1c) ↔ ∃x⟨{x}, ⟨r, a⟩⟩ ∈ ( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ))
7		oteltxp 5783	. . . . . . . . . 10 ⊢ (⟨{x}, ⟨r, a⟩⟩ ∈ ( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ) ↔ (⟨{x}, r⟩ ∈ ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ∧ ⟨{x}, a⟩ ∈ S ))
8		snex 4112	. . . . . . . . . . . . . 14 ⊢ {x} ∈ V
9	8, 2	opex 4589	. . . . . . . . . . . . 13 ⊢ ⟨{x}, r⟩ ∈ V
10	9	elcompl 3226	. . . . . . . . . . . 12 ⊢ (⟨{x}, r⟩ ∈ ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ↔ ¬ ⟨{x}, r⟩ ∈ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c))
11		elima1c 4948	. . . . . . . . . . . . . 14 ⊢ (⟨{x}, r⟩ ∈ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ↔ ∃p⟨{p}, ⟨{x}, r⟩⟩ ∈ ( SI (1st ∩ 2nd ) ⊗ S ))
12		oteltxp 5783	. . . . . . . . . . . . . . . 16 ⊢ (⟨{p}, ⟨{x}, r⟩⟩ ∈ ( SI (1st ∩ 2nd ) ⊗ S ) ↔ (⟨{p}, {x}⟩ ∈ SI (1st ∩ 2nd ) ∧ ⟨{p}, r⟩ ∈ S ))
13		vex 2863	. . . . . . . . . . . . . . . . . . 19 ⊢ p ∈ V
14		vex 2863	. . . . . . . . . . . . . . . . . . 19 ⊢ x ∈ V
15	13, 14	opsnelsi 5775	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨{p}, {x}⟩ ∈ SI (1st ∩ 2nd ) ↔ ⟨p, x⟩ ∈ (1st ∩ 2nd ))
16		elin 3220	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨p, x⟩ ∈ (1st ∩ 2nd ) ↔ (⟨p, x⟩ ∈ 1st ∧ ⟨p, x⟩ ∈ 2nd ))
17		df-br 4641	. . . . . . . . . . . . . . . . . . . 20 ⊢ (p1st x ↔ ⟨p, x⟩ ∈ 1st )
18		df-br 4641	. . . . . . . . . . . . . . . . . . . 20 ⊢ (p2nd x ↔ ⟨p, x⟩ ∈ 2nd )
19	17, 18	anbi12i 678	. . . . . . . . . . . . . . . . . . 19 ⊢ ((p1st x ∧ p2nd x) ↔ (⟨p, x⟩ ∈ 1st ∧ ⟨p, x⟩ ∈ 2nd ))
20	14, 14	op1st2nd 5791	. . . . . . . . . . . . . . . . . . 19 ⊢ ((p1st x ∧ p2nd x) ↔ p = ⟨x, x⟩)
21	16, 19, 20	3bitr2i 264	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨p, x⟩ ∈ (1st ∩ 2nd ) ↔ p = ⟨x, x⟩)
22	15, 21	bitri 240	. . . . . . . . . . . . . . . . 17 ⊢ (⟨{p}, {x}⟩ ∈ SI (1st ∩ 2nd ) ↔ p = ⟨x, x⟩)
23	13, 2	opelssetsn 4761	. . . . . . . . . . . . . . . . 17 ⊢ (⟨{p}, r⟩ ∈ S ↔ p ∈ r)
24	22, 23	anbi12i 678	. . . . . . . . . . . . . . . 16 ⊢ ((⟨{p}, {x}⟩ ∈ SI (1st ∩ 2nd ) ∧ ⟨{p}, r⟩ ∈ S ) ↔ (p = ⟨x, x⟩ ∧ p ∈ r))
25	12, 24	bitri 240	. . . . . . . . . . . . . . 15 ⊢ (⟨{p}, ⟨{x}, r⟩⟩ ∈ ( SI (1st ∩ 2nd ) ⊗ S ) ↔ (p = ⟨x, x⟩ ∧ p ∈ r))
26	25	exbii 1582	. . . . . . . . . . . . . 14 ⊢ (∃p⟨{p}, ⟨{x}, r⟩⟩ ∈ ( SI (1st ∩ 2nd ) ⊗ S ) ↔ ∃p(p = ⟨x, x⟩ ∧ p ∈ r))
27	11, 26	bitri 240	. . . . . . . . . . . . 13 ⊢ (⟨{x}, r⟩ ∈ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ↔ ∃p(p = ⟨x, x⟩ ∧ p ∈ r))
28		df-br 4641	. . . . . . . . . . . . . 14 ⊢ (xrx ↔ ⟨x, x⟩ ∈ r)
29		df-clel 2349	. . . . . . . . . . . . . 14 ⊢ (⟨x, x⟩ ∈ r ↔ ∃p(p = ⟨x, x⟩ ∧ p ∈ r))
30	28, 29	bitri 240	. . . . . . . . . . . . 13 ⊢ (xrx ↔ ∃p(p = ⟨x, x⟩ ∧ p ∈ r))
31	27, 30	bitr4i 243	. . . . . . . . . . . 12 ⊢ (⟨{x}, r⟩ ∈ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ↔ xrx)
32	10, 31	xchbinx 301	. . . . . . . . . . 11 ⊢ (⟨{x}, r⟩ ∈ ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ↔ ¬ xrx)
33	14, 3	opelssetsn 4761	. . . . . . . . . . 11 ⊢ (⟨{x}, a⟩ ∈ S ↔ x ∈ a)
34	32, 33	anbi12ci 679	. . . . . . . . . 10 ⊢ ((⟨{x}, r⟩ ∈ ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ∧ ⟨{x}, a⟩ ∈ S ) ↔ (x ∈ a ∧ ¬ xrx))
35	7, 34	bitri 240	. . . . . . . . 9 ⊢ (⟨{x}, ⟨r, a⟩⟩ ∈ ( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ) ↔ (x ∈ a ∧ ¬ xrx))
36	35	exbii 1582	. . . . . . . 8 ⊢ (∃x⟨{x}, ⟨r, a⟩⟩ ∈ ( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ) ↔ ∃x(x ∈ a ∧ ¬ xrx))
37	6, 36	bitri 240	. . . . . . 7 ⊢ (⟨r, a⟩ ∈ (( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ) “ 1c) ↔ ∃x(x ∈ a ∧ ¬ xrx))
38		df-rex 2621	. . . . . . 7 ⊢ (∃x ∈ a ¬ xrx ↔ ∃x(x ∈ a ∧ ¬ xrx))
39		rexnal 2626	. . . . . . 7 ⊢ (∃x ∈ a ¬ xrx ↔ ¬ ∀x ∈ a xrx)
40	37, 38, 39	3bitr2i 264	. . . . . 6 ⊢ (⟨r, a⟩ ∈ (( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ) “ 1c) ↔ ¬ ∀x ∈ a xrx)
41	40	con2bii 322	. . . . 5 ⊢ (∀x ∈ a xrx ↔ ¬ ⟨r, a⟩ ∈ (( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ) “ 1c))
42	5, 41	bitr4i 243	. . . 4 ⊢ (⟨r, a⟩ ∈ ∼ (( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ) “ 1c) ↔ ∀x ∈ a xrx)
43	42	opabbi2i 4867	. . 3 ⊢ ∼ (( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ) “ 1c) = {⟨r, a⟩ ∣ ∀x ∈ a xrx}
44	1, 43	eqtr4i 2376	. 2 ⊢ Ref = ∼ (( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ) “ 1c)
45		1stex 4740	. . . . . . . . . 10 ⊢ 1st ∈ V
46		2ndex 5113	. . . . . . . . . 10 ⊢ 2nd ∈ V
47	45, 46	inex 4106	. . . . . . . . 9 ⊢ (1st ∩ 2nd ) ∈ V
48	47	siex 4754	. . . . . . . 8 ⊢ SI (1st ∩ 2nd ) ∈ V
49		ssetex 4745	. . . . . . . 8 ⊢ S ∈ V
50	48, 49	txpex 5786	. . . . . . 7 ⊢ ( SI (1st ∩ 2nd ) ⊗ S ) ∈ V
51		1cex 4143	. . . . . . 7 ⊢ 1c ∈ V
52	50, 51	imaex 4748	. . . . . 6 ⊢ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ∈ V
53	52	complex 4105	. . . . 5 ⊢ ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ∈ V
54	53, 49	txpex 5786	. . . 4 ⊢ ( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ) ∈ V
55	54, 51	imaex 4748	. . 3 ⊢ (( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ) “ 1c) ∈ V
56	55	complex 4105	. 2 ⊢ ∼ (( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ) “ 1c) ∈ V
57	44, 56	eqeltri 2423	1 ⊢ Ref ∈ V

Syntax hints:  ¬ wn 3   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616  Vcvv 2860   ∼ ccompl 3206   ∩ cin 3209  {csn 3738  1_cc1c 4135  ⟨cop 4562  {copab 4623   class class class wbr 4640  1^st c1st 4718   S csset 4720   SI csi 4721   “ cima 4723  2^nd c2nd 4784   ⊗ ctxp 5736   Ref cref 5890

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-si 4729  df-cnv 4786  df-2nd 4798  df-txp 5737  df-ref 5901

This theorem is referenced by:  partialex  5918