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Theorem uninqs 25050
 Description: Class union distributes over the intersection of two subclasses of a quotient space. Compare uniin 3849. (Contributed by FL, 25-May-2007.) (Proof shortened by Mario Carneiro, 11-Jul-2014.)
Hypothesis
Ref Expression
uninqs.1
Assertion
Ref Expression
uninqs

Proof of Theorem uninqs
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uniin 3849 . . 3
21a1i 10 . 2
3 eluni2 3833 . . . . . 6
4 eluni2 3833 . . . . . 6
53, 4anbi12i 678 . . . . 5
6 elin 3360 . . . . 5
7 reeanv 2709 . . . . 5
85, 6, 73bitr4i 268 . . . 4
9 simp3l 983 . . . . . . 7
10 simp2l 981 . . . . . . . 8
11 inelcm 3511 . . . . . . . . . . 11
12113ad2ant3 978 . . . . . . . . . 10
13 uninqs.1 . . . . . . . . . . . . . 14
1413a1i 10 . . . . . . . . . . . . 13
15 simp1l 979 . . . . . . . . . . . . . 14
1615, 10sseldd 3183 . . . . . . . . . . . . 13
17 simp1r 980 . . . . . . . . . . . . . 14
18 simp2r 982 . . . . . . . . . . . . . 14
1917, 18sseldd 3183 . . . . . . . . . . . . 13
2014, 16, 19qsdisj 6738 . . . . . . . . . . . 12
2120ord 366 . . . . . . . . . . 11
2221necon1ad 2515 . . . . . . . . . 10
2312, 22mpd 14 . . . . . . . . 9
2423, 18eqeltrd 2359 . . . . . . . 8
25 elin 3360 . . . . . . . 8
2610, 24, 25sylanbrc 645 . . . . . . 7
27 elunii 3834 . . . . . . 7
289, 26, 27syl2anc 642 . . . . . 6
29283expia 1153 . . . . 5
3029rexlimdvva 2676 . . . 4
318, 30syl5bi 208 . . 3
3231ssrdv 3187 . 2
332, 32eqssd 3198 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   w3a 934   wceq 1625   wcel 1686   wne 2448  wrex 2546   cin 3153   wss 3154  c0 3457  cuni 3829   wer 6659  cqs 6661 This theorem is referenced by:  qusp  25553 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-er 6662  df-ec 6664  df-qs 6668
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