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Theorem cover2 26461
 Description: Two ways of expressing the statement "there is a cover of by elements of such that for each set in the cover, ." Note that and must be distinct. (Contributed by Jeff Madsen, 20-Jun-2010.)
Hypotheses
Ref Expression
cover2.1
cover2.2
Assertion
Ref Expression
cover2
Distinct variable groups:   ,,   ,,,   ,,
Allowed substitution hints:   ()   ()

Proof of Theorem cover2
StepHypRef Expression
1 ssrab2 3271 . . . 4
2 cover2.1 . . . . 5
32elpw2 4191 . . . 4
41, 3mpbir 200 . . 3
5 nfra1 2606 . . . . 5
6 uniss 3864 . . . . . . . . 9
71, 6ax-mp 8 . . . . . . . 8
87sseli 3189 . . . . . . 7
9 cover2.2 . . . . . . 7
108, 9syl6eleqr 2387 . . . . . 6
11 rsp 2616 . . . . . . 7
12 elunirab 3856 . . . . . . 7
1311, 12syl6ibr 218 . . . . . 6
1410, 13impbid2 195 . . . . 5
155, 14alrimi 1757 . . . 4
16 dfcleq 2290 . . . 4
1715, 16sylibr 203 . . 3
18 unieq 3852 . . . . . . 7
1918eqeq1d 2304 . . . . . 6
2019anbi1d 685 . . . . 5
21 nfrab1 2733 . . . . . . . 8
2221nfeq2 2443 . . . . . . 7
23 eleq2 2357 . . . . . . . 8
24 rabid 2729 . . . . . . . . 9
2524simprbi 450 . . . . . . . 8
2623, 25syl6bi 219 . . . . . . 7
2722, 26ralrimi 2637 . . . . . 6
2827biantrud 493 . . . . 5
2920, 28bitr4d 247 . . . 4
3029rspcev 2897 . . 3
314, 17, 30sylancr 644 . 2
32 eleq2 2357 . . . . . . . . . 10
3332biimpar 471 . . . . . . . . 9
34 eluni2 3847 . . . . . . . . 9
3533, 34sylib 188 . . . . . . . 8
36 elpwi 3646 . . . . . . . . . 10
37 r19.29r 2697 . . . . . . . . . . . 12
3837expcom 424 . . . . . . . . . . 11
39 ssrexv 3251 . . . . . . . . . . 11
4038, 39sylan9r 639 . . . . . . . . . 10
4136, 40sylan 457 . . . . . . . . 9
4241imp 418 . . . . . . . 8
4335, 42sylan2 460 . . . . . . 7
4443anassrs 629 . . . . . 6
4544ralrimiva 2639 . . . . 5
4645anasss 628 . . . 4
4746ancom2s 777 . . 3
4847rexlimiva 2675 . 2
4931, 48impbii 180 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358  wal 1530   wceq 1632   wcel 1696  wral 2556  wrex 2557  crab 2560  cvv 2801   wss 3165  cpw 3638  cuni 3843 This theorem is referenced by:  cover2g  26462 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-in 3172  df-ss 3179  df-pw 3640  df-uni 3844
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