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Theorem refex 5911
 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.
StepHypRef Expression
1 df-ref 5900 . . 3 Ref = {r, a x a xrx}
2 vex 2862 . . . . . . 7 r V
3 vex 2862 . . . . . . 7 a V
42, 3opex 4588 . . . . . 6 r, a V
54elcompl 3225 . . . . 5 (r, a ∼ (( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ) “ 1c) ↔ ¬ r, a (( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ) “ 1c))
6 elima1c 4947 . . . . . . . 8 (r, a (( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ) “ 1c) ↔ x{x}, r, a ( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ))
7 oteltxp 5782 . . . . . . . . . 10 ({x}, r, a ( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ) ↔ ({x}, r ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) {x}, a S ))
8 snex 4111 . . . . . . . . . . . . . 14 {x} V
98, 2opex 4588 . . . . . . . . . . . . 13 {x}, r V
109elcompl 3225 . . . . . . . . . . . 12 ({x}, r ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ↔ ¬ {x}, r (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c))
11 elima1c 4947 . . . . . . . . . . . . . 14 ({x}, r (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ↔ p{p}, {x}, r ( SI (1st ∩ 2nd ) ⊗ S ))
12 oteltxp 5782 . . . . . . . . . . . . . . . 16 ({p}, {x}, r ( SI (1st ∩ 2nd ) ⊗ S ) ↔ ({p}, {x} SI (1st ∩ 2nd ) {p}, r S ))
13 vex 2862 . . . . . . . . . . . . . . . . . . 19 p V
14 vex 2862 . . . . . . . . . . . . . . . . . . 19 x V
1513, 14opsnelsi 5774 . . . . . . . . . . . . . . . . . 18 ({p}, {x} SI (1st ∩ 2nd ) ↔ p, x (1st ∩ 2nd ))
16 elin 3219 . . . . . . . . . . . . . . . . . . 19 (p, x (1st ∩ 2nd ) ↔ (p, x 1st p, x 2nd ))
17 df-br 4640 . . . . . . . . . . . . . . . . . . . 20 (p1st xp, x 1st )
18 df-br 4640 . . . . . . . . . . . . . . . . . . . 20 (p2nd xp, x 2nd )
1917, 18anbi12i 678 . . . . . . . . . . . . . . . . . . 19 ((p1st x p2nd x) ↔ (p, x 1st p, x 2nd ))
2014, 14op1st2nd 5790 . . . . . . . . . . . . . . . . . . 19 ((p1st x p2nd x) ↔ p = x, x)
2116, 19, 203bitr2i 264 . . . . . . . . . . . . . . . . . 18 (p, x (1st ∩ 2nd ) ↔ p = x, x)
2215, 21bitri 240 . . . . . . . . . . . . . . . . 17 ({p}, {x} SI (1st ∩ 2nd ) ↔ p = x, x)
2313, 2opelssetsn 4760 . . . . . . . . . . . . . . . . 17 ({p}, r S p r)
2422, 23anbi12i 678 . . . . . . . . . . . . . . . 16 (({p}, {x} SI (1st ∩ 2nd ) {p}, r S ) ↔ (p = x, x p r))
2512, 24bitri 240 . . . . . . . . . . . . . . 15 ({p}, {x}, r ( SI (1st ∩ 2nd ) ⊗ S ) ↔ (p = x, x p r))
2625exbii 1582 . . . . . . . . . . . . . 14 (p{p}, {x}, r ( SI (1st ∩ 2nd ) ⊗ S ) ↔ p(p = x, x p r))
2711, 26bitri 240 . . . . . . . . . . . . 13 ({x}, r (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ↔ p(p = x, x p r))
28 df-br 4640 . . . . . . . . . . . . . 14 (xrxx, x r)
29 df-clel 2349 . . . . . . . . . . . . . 14 (x, x rp(p = x, x p r))
3028, 29bitri 240 . . . . . . . . . . . . 13 (xrxp(p = x, x p r))
3127, 30bitr4i 243 . . . . . . . . . . . 12 ({x}, r (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ↔ xrx)
3210, 31xchbinx 301 . . . . . . . . . . 11 ({x}, r ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ↔ ¬ xrx)
3314, 3opelssetsn 4760 . . . . . . . . . . 11 ({x}, a S x a)
3432, 33anbi12ci 679 . . . . . . . . . 10 (({x}, r ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) {x}, a S ) ↔ (x a ¬ xrx))
357, 34bitri 240 . . . . . . . . 9 ({x}, r, a ( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ) ↔ (x a ¬ xrx))
3635exbii 1582 . . . . . . . 8 (x{x}, r, a ( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ) ↔ x(x a ¬ xrx))
376, 36bitri 240 . . . . . . 7 (r, a (( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ) “ 1c) ↔ x(x a ¬ xrx))
38 df-rex 2620 . . . . . . 7 (x a ¬ xrxx(x a ¬ xrx))
39 rexnal 2625 . . . . . . 7 (x a ¬ xrx ↔ ¬ x a xrx)
4037, 38, 393bitr2i 264 . . . . . 6 (r, a (( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ) “ 1c) ↔ ¬ x a xrx)
4140con2bii 322 . . . . 5 (x a xrx ↔ ¬ r, a (( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ) “ 1c))
425, 41bitr4i 243 . . . 4 (r, a ∼ (( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ) “ 1c) ↔ x a xrx)
4342opabbi2i 4866 . . 3 ∼ (( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ) “ 1c) = {r, a x a xrx}
441, 43eqtr4i 2376 . 2 Ref = ∼ (( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ) “ 1c)
45 1stex 4739 . . . . . . . . . 10 1st V
46 2ndex 5112 . . . . . . . . . 10 2nd V
4745, 46inex 4105 . . . . . . . . 9 (1st ∩ 2nd ) V
4847siex 4753 . . . . . . . 8 SI (1st ∩ 2nd ) V
49 ssetex 4744 . . . . . . . 8 S V
5048, 49txpex 5785 . . . . . . 7 ( SI (1st ∩ 2nd ) ⊗ S ) V
51 1cex 4142 . . . . . . 7 1c V
5250, 51imaex 4747 . . . . . 6 (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) V
5352complex 4104 . . . . 5 ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) V
5453, 49txpex 5785 . . . 4 ( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ) V
5554, 51imaex 4747 . . 3 (( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ) “ 1c) V
5655complex 4104 . 2 ∼ (( ∼ (( SI (1st ∩ 2nd ) ⊗ S ) “ 1c) ⊗ S ) “ 1c) V
5744, 56eqeltri 2423 1 Ref V
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615  Vcvv 2859   ∼ ccompl 3205   ∩ cin 3208  {csn 3737  1cc1c 4134  ⟨cop 4561  {copab 4622   class class class wbr 4639  1st c1st 4717   S csset 4719   SI csi 4720   “ cima 4722  2nd c2nd 4783   ⊗ ctxp 5735   Ref cref 5889 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-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-cnv 4785  df-2nd 4797  df-txp 5736  df-ref 5900 This theorem is referenced by:  partialex  5917
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