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Theorem erdisj 5972
Description: Equivalence classes do not overlap. In other words, two equivalence classes are either equal or disjoint. Theorem 74 of [Suppes] p. 83. (Contributed by set.mm contributors, 15-Jun-2004.) (Revised by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
erdisj (R Er V → ([A]R = [B]R ([A]R ∩ [B]R) = ))

Proof of Theorem erdisj
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 neq0 3560 . . . 4 (¬ ([A]R ∩ [B]R) = x x ([A]R ∩ [B]R))
2 simpl 443 . . . . . . 7 ((R Er V x ([A]R ∩ [B]R)) → R Er V)
3 inss1 3475 . . . . . . . . . . 11 ([A]R ∩ [B]R) [A]R
43sseli 3269 . . . . . . . . . 10 (x ([A]R ∩ [B]R) → x [A]R)
54adantl 452 . . . . . . . . 9 ((R Er V x ([A]R ∩ [B]R)) → x [A]R)
6 ecexr 5950 . . . . . . . . 9 (x [A]RA V)
75, 6syl 15 . . . . . . . 8 ((R Er V x ([A]R ∩ [B]R)) → A V)
8 vex 2862 . . . . . . . . 9 x V
98a1i 10 . . . . . . . 8 ((R Er V x ([A]R ∩ [B]R)) → x V)
10 inss2 3476 . . . . . . . . . . 11 ([A]R ∩ [B]R) [B]R
1110sseli 3269 . . . . . . . . . 10 (x ([A]R ∩ [B]R) → x [B]R)
1211adantl 452 . . . . . . . . 9 ((R Er V x ([A]R ∩ [B]R)) → x [B]R)
13 ecexr 5950 . . . . . . . . 9 (x [B]RB V)
1412, 13syl 15 . . . . . . . 8 ((R Er V x ([A]R ∩ [B]R)) → B V)
15 elec 5964 . . . . . . . . . 10 (x [A]RARx)
164, 15sylib 188 . . . . . . . . 9 (x ([A]R ∩ [B]R) → ARx)
1716adantl 452 . . . . . . . 8 ((R Er V x ([A]R ∩ [B]R)) → ARx)
18 elec 5964 . . . . . . . . . 10 (x [B]RBRx)
1911, 18sylib 188 . . . . . . . . 9 (x ([A]R ∩ [B]R) → BRx)
2019adantl 452 . . . . . . . 8 ((R Er V x ([A]R ∩ [B]R)) → BRx)
212, 7, 9, 14, 17, 20ertr4d 5958 . . . . . . 7 ((R Er V x ([A]R ∩ [B]R)) → ARB)
222, 21erthi 5970 . . . . . 6 ((R Er V x ([A]R ∩ [B]R)) → [A]R = [B]R)
2322ex 423 . . . . 5 (R Er V → (x ([A]R ∩ [B]R) → [A]R = [B]R))
2423exlimdv 1636 . . . 4 (R Er V → (x x ([A]R ∩ [B]R) → [A]R = [B]R))
251, 24syl5bi 208 . . 3 (R Er V → (¬ ([A]R ∩ [B]R) = → [A]R = [B]R))
2625orrd 367 . 2 (R Er V → (([A]R ∩ [B]R) = [A]R = [B]R))
2726orcomd 377 1 (R Er V → ([A]R = [B]R ([A]R ∩ [B]R) = ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   wo 357   wa 358  wex 1541   = wceq 1642   wcel 1710  Vcvv 2859  cin 3208  c0 3550   class class class wbr 4639   Er cer 5898  [cec 5945
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
This theorem is referenced by:  qsdisj  5995  ncdisjeq  6148
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