New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  oqelins4 GIF version

Theorem oqelins4 5794
 Description: Ordered quadruple membership in Ins4. (Contributed by SF, 13-Feb-2015.)
Hypothesis
Ref Expression
oqelins4.4 D V
Assertion
Ref Expression
oqelins4 (A, B, C, D Ins4 RA, B, C R)

Proof of Theorem oqelins4
Dummy variables a b p x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2867 . . 3 (A, B, C, D Ins4 RA, B, C, D V)
2 opexb 4603 . . . . 5 (A, B, C, D V ↔ (A V B, C, D V))
3 opexb 4603 . . . . . 6 (B, C, D V ↔ (B V C, D V))
43anbi2i 675 . . . . 5 ((A V B, C, D V) ↔ (A V (B V C, D V)))
52, 4bitri 240 . . . 4 (A, B, C, D V ↔ (A V (B V C, D V)))
6 opexb 4603 . . . . . . 7 (C, D V ↔ (C V D V))
76simplbi 446 . . . . . 6 (C, D V → C V)
87anim2i 552 . . . . 5 ((B V C, D V) → (B V C V))
98anim2i 552 . . . 4 ((A V (B V C, D V)) → (A V (B V C V)))
105, 9sylbi 187 . . 3 (A, B, C, D V → (A V (B V C V)))
111, 10syl 15 . 2 (A, B, C, D Ins4 R → (A V (B V C V)))
12 elex 2867 . . 3 (A, B, C RA, B, C V)
13 opexb 4603 . . . 4 (A, B, C V ↔ (A V B, C V))
14 opexb 4603 . . . . 5 (B, C V ↔ (B V C V))
1514anbi2i 675 . . . 4 ((A V B, C V) ↔ (A V (B V C V)))
1613, 15bitri 240 . . 3 (A, B, C V ↔ (A V (B V C V)))
1712, 16sylib 188 . 2 (A, B, C R → (A V (B V C V)))
18 opeq1 4578 . . . . . . 7 (x = Ax, B, C, D = A, B, C, D)
1918eleq1d 2419 . . . . . 6 (x = A → (x, B, C, D Ins4 RA, B, C, D Ins4 R))
20 opeq1 4578 . . . . . . 7 (x = Ax, B, C = A, B, C)
2120eleq1d 2419 . . . . . 6 (x = A → (x, B, C RA, B, C R))
2219, 21bibi12d 312 . . . . 5 (x = A → ((x, B, C, D Ins4 Rx, B, C R) ↔ (A, B, C, D Ins4 RA, B, C R)))
2322imbi2d 307 . . . 4 (x = A → (((B V C V) → (x, B, C, D Ins4 Rx, B, C R)) ↔ ((B V C V) → (A, B, C, D Ins4 RA, B, C R))))
24 opeq1 4578 . . . . . . . 8 (y = By, z, D = B, z, D)
2524opeq2d 4585 . . . . . . 7 (y = Bx, y, z, D = x, B, z, D)
2625eleq1d 2419 . . . . . 6 (y = B → (x, y, z, D Ins4 Rx, B, z, D Ins4 R))
27 opeq1 4578 . . . . . . . 8 (y = By, z = B, z)
2827opeq2d 4585 . . . . . . 7 (y = Bx, y, z = x, B, z)
2928eleq1d 2419 . . . . . 6 (y = B → (x, y, z Rx, B, z R))
3026, 29bibi12d 312 . . . . 5 (y = B → ((x, y, z, D Ins4 Rx, y, z R) ↔ (x, B, z, D Ins4 Rx, B, z R)))
31 opeq1 4578 . . . . . . . . 9 (z = Cz, D = C, D)
3231opeq2d 4585 . . . . . . . 8 (z = CB, z, D = B, C, D)
3332opeq2d 4585 . . . . . . 7 (z = Cx, B, z, D = x, B, C, D)
3433eleq1d 2419 . . . . . 6 (z = C → (x, B, z, D Ins4 Rx, B, C, D Ins4 R))
35 opeq2 4579 . . . . . . . 8 (z = CB, z = B, C)
3635opeq2d 4585 . . . . . . 7 (z = Cx, B, z = x, B, C)
3736eleq1d 2419 . . . . . 6 (z = C → (x, B, z Rx, B, C R))
3834, 37bibi12d 312 . . . . 5 (z = C → ((x, B, z, D Ins4 Rx, B, z R) ↔ (x, B, C, D Ins4 Rx, B, C R)))
39 df-ins4 5756 . . . . . . 7 Ins4 R = ((1st ⊗ ((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))) “ R)
4039eleq2i 2417 . . . . . 6 (x, y, z, D Ins4 Rx, y, z, D ((1st ⊗ ((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))) “ R))
41 brcnv 4892 . . . . . . . . 9 (p(1st ⊗ ((1st 2nd ) ⊗ ((1st 2nd ) 2nd )))x, y, z, Dx, y, z, D(1st ⊗ ((1st 2nd ) ⊗ ((1st 2nd ) 2nd )))p)
42 brtxp 5783 . . . . . . . . . 10 (x, y, z, D(1st ⊗ ((1st 2nd ) ⊗ ((1st 2nd ) 2nd )))pab(p = a, b x, y, z, D1st a x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))b))
43 3ancoma 941 . . . . . . . . . . . . . 14 ((p = a, b x, y, z, D1st a x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))b) ↔ (x, y, z, D1st a p = a, b x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))b))
44 3anass 938 . . . . . . . . . . . . . 14 ((x, y, z, D1st a p = a, b x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))b) ↔ (x, y, z, D1st a (p = a, b x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))b)))
45 vex 2862 . . . . . . . . . . . . . . . . 17 x V
46 vex 2862 . . . . . . . . . . . . . . . . . 18 y V
47 vex 2862 . . . . . . . . . . . . . . . . . . 19 z V
48 oqelins4.4 . . . . . . . . . . . . . . . . . . 19 D V
4947, 48opex 4588 . . . . . . . . . . . . . . . . . 18 z, D V
5046, 49opex 4588 . . . . . . . . . . . . . . . . 17 y, z, D V
5145, 50opbr1st 5501 . . . . . . . . . . . . . . . 16 (x, y, z, D1st ax = a)
52 equcom 1680 . . . . . . . . . . . . . . . 16 (x = aa = x)
5351, 52bitri 240 . . . . . . . . . . . . . . 15 (x, y, z, D1st aa = x)
5453anbi1i 676 . . . . . . . . . . . . . 14 ((x, y, z, D1st a (p = a, b x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))b)) ↔ (a = x (p = a, b x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))b)))
5543, 44, 543bitri 262 . . . . . . . . . . . . 13 ((p = a, b x, y, z, D1st a x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))b) ↔ (a = x (p = a, b x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))b)))
5655exbii 1582 . . . . . . . . . . . 12 (b(p = a, b x, y, z, D1st a x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))b) ↔ b(a = x (p = a, b x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))b)))
57 19.42v 1905 . . . . . . . . . . . 12 (b(a = x (p = a, b x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))b)) ↔ (a = x b(p = a, b x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))b)))
5856, 57bitri 240 . . . . . . . . . . 11 (b(p = a, b x, y, z, D1st a x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))b) ↔ (a = x b(p = a, b x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))b)))
5958exbii 1582 . . . . . . . . . 10 (ab(p = a, b x, y, z, D1st a x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))b) ↔ a(a = x b(p = a, b x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))b)))
60 opeq1 4578 . . . . . . . . . . . . . 14 (a = xa, b = x, b)
6160eqeq2d 2364 . . . . . . . . . . . . 13 (a = x → (p = a, bp = x, b))
6261anbi1d 685 . . . . . . . . . . . 12 (a = x → ((p = a, b x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))b) ↔ (p = x, b x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))b)))
6362exbidv 1626 . . . . . . . . . . 11 (a = x → (b(p = a, b x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))b) ↔ b(p = x, b x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))b)))
6445, 63ceqsexv 2894 . . . . . . . . . 10 (a(a = x b(p = a, b x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))b)) ↔ b(p = x, b x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))b))
6542, 59, 643bitri 262 . . . . . . . . 9 (x, y, z, D(1st ⊗ ((1st 2nd ) ⊗ ((1st 2nd ) 2nd )))pb(p = x, b x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))b))
66 ancom 437 . . . . . . . . . . . 12 ((p = x, b x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))b) ↔ (x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))b p = x, b))
67 brtxp 5783 . . . . . . . . . . . . . 14 (x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))bpa(b = p, a x, y, z, D(1st 2nd )p x, y, z, D((1st 2nd ) 2nd )a))
68 3anrot 939 . . . . . . . . . . . . . . . 16 ((b = p, a x, y, z, D(1st 2nd )p x, y, z, D((1st 2nd ) 2nd )a) ↔ (x, y, z, D(1st 2nd )p x, y, z, D((1st 2nd ) 2nd )a b = p, a))
6945, 50brco2nd 5778 . . . . . . . . . . . . . . . . . 18 (x, y, z, D(1st 2nd )py, z, D1st p)
7046, 49opbr1st 5501 . . . . . . . . . . . . . . . . . 18 (y, z, D1st py = p)
71 equcom 1680 . . . . . . . . . . . . . . . . . 18 (y = pp = y)
7269, 70, 713bitri 262 . . . . . . . . . . . . . . . . 17 (x, y, z, D(1st 2nd )pp = y)
7345, 50brco2nd 5778 . . . . . . . . . . . . . . . . . 18 (x, y, z, D((1st 2nd ) 2nd )ay, z, D(1st 2nd )a)
7446, 49brco2nd 5778 . . . . . . . . . . . . . . . . . . 19 (y, z, D(1st 2nd )az, D1st a)
7547, 48opbr1st 5501 . . . . . . . . . . . . . . . . . . 19 (z, D1st az = a)
7674, 75bitri 240 . . . . . . . . . . . . . . . . . 18 (y, z, D(1st 2nd )az = a)
77 equcom 1680 . . . . . . . . . . . . . . . . . 18 (z = aa = z)
7873, 76, 773bitri 262 . . . . . . . . . . . . . . . . 17 (x, y, z, D((1st 2nd ) 2nd )aa = z)
79 biid 227 . . . . . . . . . . . . . . . . 17 (b = p, ab = p, a)
8072, 78, 793anbi123i 1140 . . . . . . . . . . . . . . . 16 ((x, y, z, D(1st 2nd )p x, y, z, D((1st 2nd ) 2nd )a b = p, a) ↔ (p = y a = z b = p, a))
8168, 80bitri 240 . . . . . . . . . . . . . . 15 ((b = p, a x, y, z, D(1st 2nd )p x, y, z, D((1st 2nd ) 2nd )a) ↔ (p = y a = z b = p, a))
82812exbii 1583 . . . . . . . . . . . . . 14 (pa(b = p, a x, y, z, D(1st 2nd )p x, y, z, D((1st 2nd ) 2nd )a) ↔ pa(p = y a = z b = p, a))
83 opeq1 4578 . . . . . . . . . . . . . . . 16 (p = yp, a = y, a)
8483eqeq2d 2364 . . . . . . . . . . . . . . 15 (p = y → (b = p, ab = y, a))
85 opeq2 4579 . . . . . . . . . . . . . . . 16 (a = zy, a = y, z)
8685eqeq2d 2364 . . . . . . . . . . . . . . 15 (a = z → (b = y, ab = y, z))
8746, 47, 84, 86ceqsex2v 2896 . . . . . . . . . . . . . 14 (pa(p = y a = z b = p, a) ↔ b = y, z)
8867, 82, 873bitri 262 . . . . . . . . . . . . 13 (x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))bb = y, z)
8988anbi1i 676 . . . . . . . . . . . 12 ((x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))b p = x, b) ↔ (b = y, z p = x, b))
9066, 89bitri 240 . . . . . . . . . . 11 ((p = x, b x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))b) ↔ (b = y, z p = x, b))
9190exbii 1582 . . . . . . . . . 10 (b(p = x, b x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))b) ↔ b(b = y, z p = x, b))
9246, 47opex 4588 . . . . . . . . . . 11 y, z V
93 opeq2 4579 . . . . . . . . . . . 12 (b = y, zx, b = x, y, z)
9493eqeq2d 2364 . . . . . . . . . . 11 (b = y, z → (p = x, bp = x, y, z))
9592, 94ceqsexv 2894 . . . . . . . . . 10 (b(b = y, z p = x, b) ↔ p = x, y, z)
9691, 95bitri 240 . . . . . . . . 9 (b(p = x, b x, y, z, D((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))b) ↔ p = x, y, z)
9741, 65, 963bitri 262 . . . . . . . 8 (p(1st ⊗ ((1st 2nd ) ⊗ ((1st 2nd ) 2nd )))x, y, z, Dp = x, y, z)
9897rexbii 2639 . . . . . . 7 (p R p(1st ⊗ ((1st 2nd ) ⊗ ((1st 2nd ) 2nd )))x, y, z, Dp R p = x, y, z)
99 elima 4754 . . . . . . 7 (x, y, z, D ((1st ⊗ ((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))) “ R) ↔ p R p(1st ⊗ ((1st 2nd ) ⊗ ((1st 2nd ) 2nd )))x, y, z, D)
100 risset 2661 . . . . . . 7 (x, y, z Rp R p = x, y, z)
10198, 99, 1003bitr4i 268 . . . . . 6 (x, y, z, D ((1st ⊗ ((1st 2nd ) ⊗ ((1st 2nd ) 2nd ))) “ R) ↔ x, y, z R)
10240, 101bitri 240 . . . . 5 (x, y, z, D Ins4 Rx, y, z R)
10330, 38, 102vtocl2g 2918 . . . 4 ((B V C V) → (x, B, C, D Ins4 Rx, B, C R))
10423, 103vtoclg 2914 . . 3 (A V → ((B V C V) → (A, B, C, D Ins4 RA, B, C R)))
105104imp 418 . 2 ((A V (B V C V)) → (A, B, C, D Ins4 RA, B, C R))
10611, 17, 105pm5.21nii 342 1 (A, B, C, D Ins4 RA, B, C R)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  Vcvv 2859  ⟨cop 4561   class class class wbr 4639  1st c1st 4717   ∘ ccom 4721   “ cima 4722  ◡ccnv 4771  2nd c2nd 4783   ⊗ ctxp 5735   Ins4 cins4 5755 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-co 4726  df-ima 4727  df-cnv 4785  df-2nd 4797  df-txp 5736  df-ins4 5756 This theorem is referenced by:  composeex  5820  addcfnex  5824  funsex  5828  crossex  5850  domfnex  5870  ranfnex  5871  transex  5910  antisymex  5912  connexex  5913  foundex  5914  extex  5915  symex  5916  ovmuc  6130  mucex  6133  ovcelem1  6171  ceex  6174
 Copyright terms: Public domain W3C validator