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Theorem swoer 6457
 Description: Incomparability under a strict weak partial order is an equivalence relation. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
swoer.1
swoer.2
swoer.3
Assertion
Ref Expression
swoer
Distinct variable groups:   ,,,   ,,,   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem swoer
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 swoer.1 . . . . 5
2 difss 3202 . . . . 5
31, 2eqsstri 3129 . . . 4
4 relxp 4648 . . . 4
5 relss 4626 . . . 4
63, 4, 5mp2 16 . . 3
76a1i 9 . 2
8 simpr 109 . . 3
9 orcom 717 . . . . . 6
109a1i 9 . . . . 5
1110notbid 656 . . . 4
123ssbri 3972 . . . . . . 7
1312adantl 275 . . . . . 6
14 brxp 4570 . . . . . 6
1513, 14sylib 121 . . . . 5
161brdifun 6456 . . . . 5
1715, 16syl 14 . . . 4
1815simprd 113 . . . . 5
1915simpld 111 . . . . 5
201brdifun 6456 . . . . 5
2118, 19, 20syl2anc 408 . . . 4
2211, 17, 213bitr4d 219 . . 3
238, 22mpbid 146 . 2
24 simprl 520 . . . . 5
2512ad2antrl 481 . . . . . . 7
2614simplbi 272 . . . . . . 7
2725, 26syl 14 . . . . . 6
2814simprbi 273 . . . . . . 7
2925, 28syl 14 . . . . . 6
3027, 29, 16syl2anc 408 . . . . 5
3124, 30mpbid 146 . . . 4
32 simprr 521 . . . . 5
333brel 4591 . . . . . . . 8
3433simprd 113 . . . . . . 7
3532, 34syl 14 . . . . . 6
361brdifun 6456 . . . . . 6
3729, 35, 36syl2anc 408 . . . . 5
3832, 37mpbid 146 . . . 4
39 simpl 108 . . . . . . 7
40 swoer.3 . . . . . . . 8
4140swopolem 4227 . . . . . . 7
4239, 27, 35, 29, 41syl13anc 1218 . . . . . 6
4340swopolem 4227 . . . . . . . 8
4439, 35, 27, 29, 43syl13anc 1218 . . . . . . 7
45 orcom 717 . . . . . . 7
4644, 45syl6ibr 161 . . . . . 6
4742, 46orim12d 775 . . . . 5
48 or4 760 . . . . 5
4947, 48syl6ib 160 . . . 4
5031, 38, 49mtord 772 . . 3
511brdifun 6456 . . . 4
5227, 35, 51syl2anc 408 . . 3
5350, 52mpbird 166 . 2
54 swoer.2 . . . . . . 7
5554, 40swopo 4228 . . . . . 6
56 poirr 4229 . . . . . 6
5755, 56sylan 281 . . . . 5
58 pm1.2 745 . . . . 5
5957, 58nsyl 617 . . . 4
60 simpr 109 . . . . 5
611brdifun 6456 . . . . 5
6260, 60, 61syl2anc 408 . . . 4
6359, 62mpbird 166 . . 3
643ssbri 3972 . . . . 5
65 brxp 4570 . . . . . 6
6665simplbi 272 . . . . 5
6764, 66syl 14 . . . 4