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

Theorem qsdisj 5995
 Description: Members of a quotient set do not overlap. (Contributed by Rodolfo Medina, 12-Oct-2010.) (Revised by Mario Carneiro, 11-Jul-2014.)
Hypotheses
Ref Expression
qsdisj.1 (φR Er V)
qsdisj.2 (φB (A / R))
qsdisj.3 (φC (A / R))
Assertion
Ref Expression
qsdisj (φ → (B = C (BC) = ))

Proof of Theorem qsdisj
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 qsdisj.2 . 2 (φB (A / R))
2 eqid 2353 . . 3 (A / R) = (A / R)
3 eqeq1 2359 . . . 4 ([x]R = B → ([x]R = CB = C))
4 ineq1 3450 . . . . 5 ([x]R = B → ([x]RC) = (BC))
54eqeq1d 2361 . . . 4 ([x]R = B → (([x]RC) = ↔ (BC) = ))
63, 5orbi12d 690 . . 3 ([x]R = B → (([x]R = C ([x]RC) = ) ↔ (B = C (BC) = )))
7 qsdisj.3 . . . . 5 (φC (A / R))
87adantr 451 . . . 4 ((φ x A) → C (A / R))
9 eqeq2 2362 . . . . . 6 ([y]R = C → ([x]R = [y]R ↔ [x]R = C))
10 ineq2 3451 . . . . . . 7 ([y]R = C → ([x]R ∩ [y]R) = ([x]RC))
1110eqeq1d 2361 . . . . . 6 ([y]R = C → (([x]R ∩ [y]R) = ↔ ([x]RC) = ))
129, 11orbi12d 690 . . . . 5 ([y]R = C → (([x]R = [y]R ([x]R ∩ [y]R) = ) ↔ ([x]R = C ([x]RC) = )))
13 qsdisj.1 . . . . . . 7 (φR Er V)
1413ad2antrr 706 . . . . . 6 (((φ x A) y A) → R Er V)
15 erdisj 5972 . . . . . 6 (R Er V → ([x]R = [y]R ([x]R ∩ [y]R) = ))
1614, 15syl 15 . . . . 5 (((φ x A) y A) → ([x]R = [y]R ([x]R ∩ [y]R) = ))
172, 12, 16ectocld 5991 . . . 4 (((φ x A) C (A / R)) → ([x]R = C ([x]RC) = ))
188, 17mpdan 649 . . 3 ((φ x A) → ([x]R = C ([x]RC) = ))
192, 6, 18ectocld 5991 . 2 ((φ B (A / R)) → (B = C (BC) = ))
201, 19mpdan 649 1 (φ → (B = C (BC) = ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∩ cin 3208  ∅c0 3550   class class class wbr 4639   Er cer 5898  [cec 5945   / cqs 5946 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-ima 4727  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator