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Theorem qsdisj 6981
 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
qsdisj.2
qsdisj.3
Assertion
Ref Expression
qsdisj

Proof of Theorem qsdisj
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 qsdisj.2 . 2
2 eqid 2436 . . 3
3 eqeq1 2442 . . . 4
4 ineq1 3535 . . . . 5
54eqeq1d 2444 . . . 4
63, 5orbi12d 691 . . 3
7 qsdisj.3 . . . . 5
87adantr 452 . . . 4
9 eqeq2 2445 . . . . . 6
10 ineq2 3536 . . . . . . 7
1110eqeq1d 2444 . . . . . 6
129, 11orbi12d 691 . . . . 5
13 qsdisj.1 . . . . . . 7
1413ad2antrr 707 . . . . . 6
15 erdisj 6952 . . . . . 6
1614, 15syl 16 . . . . 5
172, 12, 16ectocld 6971 . . . 4
188, 17mpdan 650 . . 3
192, 6, 18ectocld 6971 . 2
201, 19mpdan 650 1
 Colors of variables: wff set class Syntax hints:   wi 4   wo 358   wa 359   wceq 1652   wcel 1725   cin 3319  c0 3628   wer 6902  cec 6903  cqs 6904 This theorem is referenced by:  qsdisj2  6982  uniinqs  6984  cldsubg  18140  erprt  26722 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-er 6905  df-ec 6907  df-qs 6911
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