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Theorem swopo 4034
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 2395 . . . 4 (φy A z A (y𝑅z → ¬ z𝑅y))
5 breq1 3758 . . . . . 6 (y = x → (y𝑅zx𝑅z))
6 breq2 3759 . . . . . . 7 (y = x → (z𝑅yz𝑅x))
76notbid 591 . . . . . 6 (y = x → (¬ z𝑅y ↔ ¬ z𝑅x))
85, 7imbi12d 223 . . . . 5 (y = x → ((y𝑅z → ¬ z𝑅y) ↔ (x𝑅z → ¬ z𝑅x)))
9 breq2 3759 . . . . . 6 (z = x → (x𝑅zx𝑅x))
10 breq1 3758 . . . . . . 7 (z = x → (z𝑅xx𝑅x))
1110notbid 591 . . . . . 6 (z = x → (¬ z𝑅x ↔ ¬ x𝑅x))
129, 11imbi12d 223 . . . . 5 (z = x → ((x𝑅z → ¬ z𝑅x) ↔ (x𝑅x → ¬ x𝑅x)))
138, 12rspc2va 2657 . . . 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 548 . 2 ((φ x A) → ¬ x𝑅x)
1633adantr1 1062 . . 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 647 . . . . 5 (((φ (x A y A z A)) x𝑅y) → (z𝑅y x𝑅z))
2019ord 642 . . . 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 4032 1 (φ𝑅 Po A)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   wo 628   w3a 884   wcel 1390  wral 2300   class class class wbr 3755   Po wpo 4022
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-po 4024
This theorem is referenced by:  swoer  6070
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