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Theorem cover2 26415
 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 3428 . . . 4
2 cover2.1 . . . . 5
32elpw2 4364 . . . 4
41, 3mpbir 201 . . 3
5 nfra1 2756 . . . . 5
61unissi 4038 . . . . . . . 8
76sseli 3344 . . . . . . 7
8 cover2.2 . . . . . . 7
97, 8syl6eleqr 2527 . . . . . 6
10 rsp 2766 . . . . . . 7
11 elunirab 4028 . . . . . . 7
1210, 11syl6ibr 219 . . . . . 6
139, 12impbid2 196 . . . . 5
145, 13alrimi 1781 . . . 4
15 dfcleq 2430 . . . 4
1614, 15sylibr 204 . . 3
17 unieq 4024 . . . . . . 7
1817eqeq1d 2444 . . . . . 6
1918anbi1d 686 . . . . 5
20 nfrab1 2888 . . . . . . . 8
2120nfeq2 2583 . . . . . . 7
22 eleq2 2497 . . . . . . . 8
23 rabid 2884 . . . . . . . . 9
2423simprbi 451 . . . . . . . 8
2522, 24syl6bi 220 . . . . . . 7
2621, 25ralrimi 2787 . . . . . 6
2726biantrud 494 . . . . 5
2819, 27bitr4d 248 . . . 4
2928rspcev 3052 . . 3
304, 16, 29sylancr 645 . 2
31 eleq2 2497 . . . . . . . . . 10
3231biimpar 472 . . . . . . . . 9
33 eluni2 4019 . . . . . . . . 9
3432, 33sylib 189 . . . . . . . 8
35 elpwi 3807 . . . . . . . . . 10
36 r19.29r 2847 . . . . . . . . . . . 12
3736expcom 425 . . . . . . . . . . 11
38 ssrexv 3408 . . . . . . . . . . 11
3937, 38sylan9r 640 . . . . . . . . . 10
4035, 39sylan 458 . . . . . . . . 9
4140imp 419 . . . . . . . 8
4234, 41sylan2 461 . . . . . . 7
4342anassrs 630 . . . . . 6
4443ralrimiva 2789 . . . . 5
4544anasss 629 . . . 4
4645ancom2s 778 . . 3
4746rexlimiva 2825 . 2
4830, 47impbii 181 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359  wal 1549   wceq 1652   wcel 1725  wral 2705  wrex 2706  crab 2709  cvv 2956   wss 3320  cpw 3799  cuni 4015 This theorem is referenced by:  cover2g  26416 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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-in 3327  df-ss 3334  df-pw 3801  df-uni 4016
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