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Theorem swopo 4017
Description: A strict weak order is a partial order. (Contributed by Mario Carneiro, 9-Jul-2014.)
Hypotheses
Ref Expression
swopo.1 ((φ (y A z A)) → (y𝑅z → ¬ z𝑅y))
swopo.2 ((φ (x A y A z A)) → (x𝑅y → (x𝑅z z𝑅y)))
Assertion
Ref Expression
swopo (φ𝑅 Po A)
Distinct variable groups:   x,y,z,A   x,𝑅,y,z   φ,x,y,z

Proof of Theorem swopo
StepHypRef Expression
1 id 19 . . . . 5 (x Ax A)
21ancli 306 . . . 4 (x A → (x A x A))
3 swopo.1 . . . . 5 ((φ (y A z A)) → (y𝑅z → ¬ z𝑅y))
43ralrimivva 2379 . . . 4 (φy A z A (y𝑅z → ¬ z𝑅y))
5 breq1 3741 . . . . . 6 (y = x → (y𝑅zx𝑅z))
6 breq2 3742 . . . . . . 7 (y = x → (z𝑅yz𝑅x))
76notbid 579 . . . . . 6 (y = x → (¬ z𝑅y ↔ ¬ z𝑅x))
85, 7imbi12d 223 . . . . 5 (y = x → ((y𝑅z → ¬ z𝑅y) ↔ (x𝑅z → ¬ z𝑅x)))
9 breq2 3742 . . . . . 6 (z = x → (x𝑅zx𝑅x))
10 breq1 3741 . . . . . . 7 (z = x → (z𝑅xx𝑅x))
1110notbid 579 . . . . . 6 (z = x → (¬ z𝑅x ↔ ¬ x𝑅x))
129, 11imbi12d 223 . . . . 5 (z = x → ((x𝑅z → ¬ z𝑅x) ↔ (x𝑅x → ¬ x𝑅x)))
138, 12rspc2va 2640 . . . 4 (((x A x A) y A z A (y𝑅z → ¬ z𝑅y)) → (x𝑅x → ¬ x𝑅x))
142, 4, 13syl2anr 274 . . 3 ((φ x A) → (x𝑅x → ¬ x𝑅x))
1514pm2.01d 536 . 2 ((φ x A) → ¬ x𝑅x)
1633adantr1 1051 . . 3 ((φ (x A y A z A)) → (y𝑅z → ¬ z𝑅y))
17 swopo.2 . . . . . . 7 ((φ (x A y A z A)) → (x𝑅y → (x𝑅z z𝑅y)))
1817imp 115 . . . . . 6 (((φ (x A y A z A)) x𝑅y) → (x𝑅z z𝑅y))
1918orcomd 635 . . . . 5 (((φ (x A y A z A)) x𝑅y) → (z𝑅y x𝑅z))
2019ord 630 . . . 4 (((φ (x A y A z A)) x𝑅y) → (¬ z𝑅yx𝑅z))
2120expimpd 345 . . 3 ((φ (x A y A z A)) → ((x𝑅y ¬ z𝑅y) → x𝑅z))
2216, 21sylan2d 278 . 2 ((φ (x A y A z A)) → ((x𝑅y y𝑅z) → x𝑅z))
2315, 22ispod 4015 1 (φ𝑅 Po A)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   wo 616   w3a 873   wcel 1374  wral 2284   class class class wbr 3738   Po wpo 4005
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-v 2537  df-un 2899  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-po 4007
This theorem is referenced by:  swoer  6045
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