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Theorem uniinqs 7448
 Description: Class union distributes over the intersection of two subclasses of a quotient space. Compare uniin 4221. (Contributed by FL, 25-May-2007.) (Proof shortened by Mario Carneiro, 11-Jul-2014.)
Hypothesis
Ref Expression
uniinqs.1
Assertion
Ref Expression
uniinqs

Proof of Theorem uniinqs
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uniin 4221 . . 3
21a1i 11 . 2
3 eluni2 4205 . . . . . 6
4 eluni2 4205 . . . . . 6
53, 4anbi12i 704 . . . . 5
6 elin 3619 . . . . 5
7 reeanv 2960 . . . . 5
85, 6, 73bitr4i 281 . . . 4
9 simp3l 1037 . . . . . . 7
10 simp2l 1035 . . . . . . . 8
11 inelcm 3821 . . . . . . . . . . 11
12113ad2ant3 1032 . . . . . . . . . 10
13 uniinqs.1 . . . . . . . . . . . . . 14
1413a1i 11 . . . . . . . . . . . . 13
15 simp1l 1033 . . . . . . . . . . . . . 14
1615, 10sseldd 3435 . . . . . . . . . . . . 13
17 simp1r 1034 . . . . . . . . . . . . . 14
18 simp2r 1036 . . . . . . . . . . . . . 14
1917, 18sseldd 3435 . . . . . . . . . . . . 13
2014, 16, 19qsdisj 7445 . . . . . . . . . . . 12
2120ord 379 . . . . . . . . . . 11
2221necon1ad 2643 . . . . . . . . . 10
2312, 22mpd 15 . . . . . . . . 9
2423, 18eqeltrd 2531 . . . . . . . 8
2510, 24elind 3620 . . . . . . 7
26 elunii 4206 . . . . . . 7
279, 25, 26syl2anc 667 . . . . . 6
28273expia 1211 . . . . 5
2928rexlimdvva 2888 . . . 4
308, 29syl5bi 221 . . 3
3130ssrdv 3440 . 2
322, 31eqssd 3451 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 371   w3a 986   wceq 1446   wcel 1889   wne 2624  wrex 2740   cin 3405   wss 3406  c0 3733  cuni 4201   wer 7365  cqs 7367 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-er 7368  df-ec 7370  df-qs 7374 This theorem is referenced by: (None)
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