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Theorem swopo 4815
Description: A strict weak order is a partial order. (Contributed by Mario Carneiro, 9-Jul-2014.)
Hypotheses
Ref Expression
swopo.1
swopo.2
Assertion
Ref Expression
swopo
Distinct variable groups:   , , ,   , , ,   , , ,

Proof of Theorem swopo
StepHypRef Expression
1 id 22 . . . . 5
21ancli 551 . . . 4
3 swopo.1 . . . . 5
43ralrimivva 2878 . . . 4
5 breq1 4455 . . . . . 6
6 breq2 4456 . . . . . . 7
76notbid 294 . . . . . 6
85, 7imbi12d 320 . . . . 5
9 breq2 4456 . . . . . 6
10 breq1 4455 . . . . . . 7
1110notbid 294 . . . . . 6
129, 11imbi12d 320 . . . . 5
138, 12rspc2va 3220 . . . 4
142, 4, 13syl2anr 478 . . 3
1514pm2.01d 169 . 2
1633adantr1 1155 . . 3
17 swopo.2 . . . . . . 7
1817imp 429 . . . . . 6
1918orcomd 388 . . . . 5
2019ord 377 . . . 4
2120expimpd 603 . . 3
2216, 21sylan2d 482 . 2
2315, 22ispod 4813 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  \/wo 368  /\wa 369  /\w3a 973  e.wcel 1818  A.wral 2807   class class class wbr 4452  Powpo 4803
This theorem is referenced by:  swoer  7358  swoso  7361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-po 4805
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