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Theorem pw1disj 4167
 Description: Two unit power classes are disjoint iff the classes themselves are disjoint. (Contributed by SF, 26-Jan-2015.)
Assertion
Ref Expression
pw1disj 1 1

Proof of Theorem pw1disj
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 disj 3591 . . . . . 6 1 1 1 1
2 eleq1 2413 . . . . . . . 8 1 1
32notbid 285 . . . . . . 7 1 1
43rspccv 2952 . . . . . 6 1 1 1 1
51, 4sylbi 187 . . . . 5 1 1 1 1
6 snelpw1 4146 . . . . 5 1
7 snelpw1 4146 . . . . . 6 1
87notbii 287 . . . . 5 1
95, 6, 83imtr3g 260 . . . 4 1 1
109ralrimiv 2696 . . 3 1 1
11 disj 3591 . . 3
1210, 11sylibr 203 . 2 1 1
13 elpw1 4144 . . . . 5 1
14 disj 3591 . . . . . . . . 9
15 rsp 2674 . . . . . . . . 9
1614, 15sylbi 187 . . . . . . . 8
1716imp 418 . . . . . . 7
18 eleq1 2413 . . . . . . . . 9 1 1
19 snelpw1 4146 . . . . . . . . 9 1
2018, 19syl6bb 252 . . . . . . . 8 1
2120notbid 285 . . . . . . 7 1
2217, 21syl5ibrcom 213 . . . . . 6 1
2322rexlimdva 2738 . . . . 5 1
2413, 23syl5bi 208 . . . 4 1 1
2524ralrimiv 2696 . . 3 1 1
26 disj 3591 . . 3 1 1 1 1
2725, 26sylibr 203 . 2 1 1
2812, 27impbii 180 1 1 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 176   wa 358   wceq 1642   wcel 1710  wral 2614  wrex 2615   cin 3208  c0 3550  csn 3737  1 cpw1 4135 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-pw 3724  df-sn 3741  df-1c 4136  df-pw1 4137 This theorem is referenced by: (None)
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