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Theorem isopos 33986
Description: The predicate "is an orthoposet." (Contributed by NM, 20-Oct-2011.) (Revised by NM, 14-Sep-2018.)
Hypotheses
Ref Expression
isopos.b 𝐵 = (Base‘𝐾)
isopos.e 𝑈 = (lub‘𝐾)
isopos.g 𝐺 = (glb‘𝐾)
isopos.l = (le‘𝐾)
isopos.o = (oc‘𝐾)
isopos.j = (join‘𝐾)
isopos.m = (meet‘𝐾)
isopos.f 0 = (0.‘𝐾)
isopos.u 1 = (1.‘𝐾)
Assertion
Ref Expression
isopos (𝐾 ∈ OP ↔ ((𝐾 ∈ Poset ∧ 𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺) ∧ ∀𝑥𝐵𝑦𝐵 ((( 𝑥) ∈ 𝐵 ∧ ( ‘( 𝑥)) = 𝑥 ∧ (𝑥 𝑦 → ( 𝑦) ( 𝑥))) ∧ (𝑥 ( 𝑥)) = 1 ∧ (𝑥 ( 𝑥)) = 0 )))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥, ,𝑦   𝑥,𝐾,𝑦
Allowed substitution hints:   𝑈(𝑥,𝑦)   1 (𝑥,𝑦)   𝐺(𝑥,𝑦)   (𝑥,𝑦)   (𝑥,𝑦)   (𝑥,𝑦)   0 (𝑥,𝑦)

Proof of Theorem isopos
Dummy variables 𝑛 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6158 . . . . . . 7 (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
2 isopos.b . . . . . . 7 𝐵 = (Base‘𝐾)
31, 2syl6eqr 2673 . . . . . 6 (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
4 fveq2 6158 . . . . . . . 8 (𝑝 = 𝐾 → (lub‘𝑝) = (lub‘𝐾))
5 isopos.e . . . . . . . 8 𝑈 = (lub‘𝐾)
64, 5syl6eqr 2673 . . . . . . 7 (𝑝 = 𝐾 → (lub‘𝑝) = 𝑈)
76dmeqd 5296 . . . . . 6 (𝑝 = 𝐾 → dom (lub‘𝑝) = dom 𝑈)
83, 7eleq12d 2692 . . . . 5 (𝑝 = 𝐾 → ((Base‘𝑝) ∈ dom (lub‘𝑝) ↔ 𝐵 ∈ dom 𝑈))
9 fveq2 6158 . . . . . . . 8 (𝑝 = 𝐾 → (glb‘𝑝) = (glb‘𝐾))
10 isopos.g . . . . . . . 8 𝐺 = (glb‘𝐾)
119, 10syl6eqr 2673 . . . . . . 7 (𝑝 = 𝐾 → (glb‘𝑝) = 𝐺)
1211dmeqd 5296 . . . . . 6 (𝑝 = 𝐾 → dom (glb‘𝑝) = dom 𝐺)
133, 12eleq12d 2692 . . . . 5 (𝑝 = 𝐾 → ((Base‘𝑝) ∈ dom (glb‘𝑝) ↔ 𝐵 ∈ dom 𝐺))
148, 13anbi12d 746 . . . 4 (𝑝 = 𝐾 → (((Base‘𝑝) ∈ dom (lub‘𝑝) ∧ (Base‘𝑝) ∈ dom (glb‘𝑝)) ↔ (𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺)))
15 fveq2 6158 . . . . . . . 8 (𝑝 = 𝐾 → (oc‘𝑝) = (oc‘𝐾))
16 isopos.o . . . . . . . 8 = (oc‘𝐾)
1715, 16syl6eqr 2673 . . . . . . 7 (𝑝 = 𝐾 → (oc‘𝑝) = )
1817eqeq2d 2631 . . . . . 6 (𝑝 = 𝐾 → (𝑛 = (oc‘𝑝) ↔ 𝑛 = ))
193eleq2d 2684 . . . . . . . . . 10 (𝑝 = 𝐾 → ((𝑛𝑥) ∈ (Base‘𝑝) ↔ (𝑛𝑥) ∈ 𝐵))
20 fveq2 6158 . . . . . . . . . . . . 13 (𝑝 = 𝐾 → (le‘𝑝) = (le‘𝐾))
21 isopos.l . . . . . . . . . . . . 13 = (le‘𝐾)
2220, 21syl6eqr 2673 . . . . . . . . . . . 12 (𝑝 = 𝐾 → (le‘𝑝) = )
2322breqd 4634 . . . . . . . . . . 11 (𝑝 = 𝐾 → (𝑥(le‘𝑝)𝑦𝑥 𝑦))
2422breqd 4634 . . . . . . . . . . 11 (𝑝 = 𝐾 → ((𝑛𝑦)(le‘𝑝)(𝑛𝑥) ↔ (𝑛𝑦) (𝑛𝑥)))
2523, 24imbi12d 334 . . . . . . . . . 10 (𝑝 = 𝐾 → ((𝑥(le‘𝑝)𝑦 → (𝑛𝑦)(le‘𝑝)(𝑛𝑥)) ↔ (𝑥 𝑦 → (𝑛𝑦) (𝑛𝑥))))
2619, 253anbi13d 1398 . . . . . . . . 9 (𝑝 = 𝐾 → (((𝑛𝑥) ∈ (Base‘𝑝) ∧ (𝑛‘(𝑛𝑥)) = 𝑥 ∧ (𝑥(le‘𝑝)𝑦 → (𝑛𝑦)(le‘𝑝)(𝑛𝑥))) ↔ ((𝑛𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛𝑥)) = 𝑥 ∧ (𝑥 𝑦 → (𝑛𝑦) (𝑛𝑥)))))
27 fveq2 6158 . . . . . . . . . . . 12 (𝑝 = 𝐾 → (join‘𝑝) = (join‘𝐾))
28 isopos.j . . . . . . . . . . . 12 = (join‘𝐾)
2927, 28syl6eqr 2673 . . . . . . . . . . 11 (𝑝 = 𝐾 → (join‘𝑝) = )
3029oveqd 6632 . . . . . . . . . 10 (𝑝 = 𝐾 → (𝑥(join‘𝑝)(𝑛𝑥)) = (𝑥 (𝑛𝑥)))
31 fveq2 6158 . . . . . . . . . . 11 (𝑝 = 𝐾 → (1.‘𝑝) = (1.‘𝐾))
32 isopos.u . . . . . . . . . . 11 1 = (1.‘𝐾)
3331, 32syl6eqr 2673 . . . . . . . . . 10 (𝑝 = 𝐾 → (1.‘𝑝) = 1 )
3430, 33eqeq12d 2636 . . . . . . . . 9 (𝑝 = 𝐾 → ((𝑥(join‘𝑝)(𝑛𝑥)) = (1.‘𝑝) ↔ (𝑥 (𝑛𝑥)) = 1 ))
35 fveq2 6158 . . . . . . . . . . . 12 (𝑝 = 𝐾 → (meet‘𝑝) = (meet‘𝐾))
36 isopos.m . . . . . . . . . . . 12 = (meet‘𝐾)
3735, 36syl6eqr 2673 . . . . . . . . . . 11 (𝑝 = 𝐾 → (meet‘𝑝) = )
3837oveqd 6632 . . . . . . . . . 10 (𝑝 = 𝐾 → (𝑥(meet‘𝑝)(𝑛𝑥)) = (𝑥 (𝑛𝑥)))
39 fveq2 6158 . . . . . . . . . . 11 (𝑝 = 𝐾 → (0.‘𝑝) = (0.‘𝐾))
40 isopos.f . . . . . . . . . . 11 0 = (0.‘𝐾)
4139, 40syl6eqr 2673 . . . . . . . . . 10 (𝑝 = 𝐾 → (0.‘𝑝) = 0 )
4238, 41eqeq12d 2636 . . . . . . . . 9 (𝑝 = 𝐾 → ((𝑥(meet‘𝑝)(𝑛𝑥)) = (0.‘𝑝) ↔ (𝑥 (𝑛𝑥)) = 0 ))
4326, 34, 423anbi123d 1396 . . . . . . . 8 (𝑝 = 𝐾 → ((((𝑛𝑥) ∈ (Base‘𝑝) ∧ (𝑛‘(𝑛𝑥)) = 𝑥 ∧ (𝑥(le‘𝑝)𝑦 → (𝑛𝑦)(le‘𝑝)(𝑛𝑥))) ∧ (𝑥(join‘𝑝)(𝑛𝑥)) = (1.‘𝑝) ∧ (𝑥(meet‘𝑝)(𝑛𝑥)) = (0.‘𝑝)) ↔ (((𝑛𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛𝑥)) = 𝑥 ∧ (𝑥 𝑦 → (𝑛𝑦) (𝑛𝑥))) ∧ (𝑥 (𝑛𝑥)) = 1 ∧ (𝑥 (𝑛𝑥)) = 0 )))
443, 43raleqbidv 3145 . . . . . . 7 (𝑝 = 𝐾 → (∀𝑦 ∈ (Base‘𝑝)(((𝑛𝑥) ∈ (Base‘𝑝) ∧ (𝑛‘(𝑛𝑥)) = 𝑥 ∧ (𝑥(le‘𝑝)𝑦 → (𝑛𝑦)(le‘𝑝)(𝑛𝑥))) ∧ (𝑥(join‘𝑝)(𝑛𝑥)) = (1.‘𝑝) ∧ (𝑥(meet‘𝑝)(𝑛𝑥)) = (0.‘𝑝)) ↔ ∀𝑦𝐵 (((𝑛𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛𝑥)) = 𝑥 ∧ (𝑥 𝑦 → (𝑛𝑦) (𝑛𝑥))) ∧ (𝑥 (𝑛𝑥)) = 1 ∧ (𝑥 (𝑛𝑥)) = 0 )))
453, 44raleqbidv 3145 . . . . . 6 (𝑝 = 𝐾 → (∀𝑥 ∈ (Base‘𝑝)∀𝑦 ∈ (Base‘𝑝)(((𝑛𝑥) ∈ (Base‘𝑝) ∧ (𝑛‘(𝑛𝑥)) = 𝑥 ∧ (𝑥(le‘𝑝)𝑦 → (𝑛𝑦)(le‘𝑝)(𝑛𝑥))) ∧ (𝑥(join‘𝑝)(𝑛𝑥)) = (1.‘𝑝) ∧ (𝑥(meet‘𝑝)(𝑛𝑥)) = (0.‘𝑝)) ↔ ∀𝑥𝐵𝑦𝐵 (((𝑛𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛𝑥)) = 𝑥 ∧ (𝑥 𝑦 → (𝑛𝑦) (𝑛𝑥))) ∧ (𝑥 (𝑛𝑥)) = 1 ∧ (𝑥 (𝑛𝑥)) = 0 )))
4618, 45anbi12d 746 . . . . 5 (𝑝 = 𝐾 → ((𝑛 = (oc‘𝑝) ∧ ∀𝑥 ∈ (Base‘𝑝)∀𝑦 ∈ (Base‘𝑝)(((𝑛𝑥) ∈ (Base‘𝑝) ∧ (𝑛‘(𝑛𝑥)) = 𝑥 ∧ (𝑥(le‘𝑝)𝑦 → (𝑛𝑦)(le‘𝑝)(𝑛𝑥))) ∧ (𝑥(join‘𝑝)(𝑛𝑥)) = (1.‘𝑝) ∧ (𝑥(meet‘𝑝)(𝑛𝑥)) = (0.‘𝑝))) ↔ (𝑛 = ∧ ∀𝑥𝐵𝑦𝐵 (((𝑛𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛𝑥)) = 𝑥 ∧ (𝑥 𝑦 → (𝑛𝑦) (𝑛𝑥))) ∧ (𝑥 (𝑛𝑥)) = 1 ∧ (𝑥 (𝑛𝑥)) = 0 ))))
4746exbidv 1847 . . . 4 (𝑝 = 𝐾 → (∃𝑛(𝑛 = (oc‘𝑝) ∧ ∀𝑥 ∈ (Base‘𝑝)∀𝑦 ∈ (Base‘𝑝)(((𝑛𝑥) ∈ (Base‘𝑝) ∧ (𝑛‘(𝑛𝑥)) = 𝑥 ∧ (𝑥(le‘𝑝)𝑦 → (𝑛𝑦)(le‘𝑝)(𝑛𝑥))) ∧ (𝑥(join‘𝑝)(𝑛𝑥)) = (1.‘𝑝) ∧ (𝑥(meet‘𝑝)(𝑛𝑥)) = (0.‘𝑝))) ↔ ∃𝑛(𝑛 = ∧ ∀𝑥𝐵𝑦𝐵 (((𝑛𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛𝑥)) = 𝑥 ∧ (𝑥 𝑦 → (𝑛𝑦) (𝑛𝑥))) ∧ (𝑥 (𝑛𝑥)) = 1 ∧ (𝑥 (𝑛𝑥)) = 0 ))))
4814, 47anbi12d 746 . . 3 (𝑝 = 𝐾 → ((((Base‘𝑝) ∈ dom (lub‘𝑝) ∧ (Base‘𝑝) ∈ dom (glb‘𝑝)) ∧ ∃𝑛(𝑛 = (oc‘𝑝) ∧ ∀𝑥 ∈ (Base‘𝑝)∀𝑦 ∈ (Base‘𝑝)(((𝑛𝑥) ∈ (Base‘𝑝) ∧ (𝑛‘(𝑛𝑥)) = 𝑥 ∧ (𝑥(le‘𝑝)𝑦 → (𝑛𝑦)(le‘𝑝)(𝑛𝑥))) ∧ (𝑥(join‘𝑝)(𝑛𝑥)) = (1.‘𝑝) ∧ (𝑥(meet‘𝑝)(𝑛𝑥)) = (0.‘𝑝)))) ↔ ((𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺) ∧ ∃𝑛(𝑛 = ∧ ∀𝑥𝐵𝑦𝐵 (((𝑛𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛𝑥)) = 𝑥 ∧ (𝑥 𝑦 → (𝑛𝑦) (𝑛𝑥))) ∧ (𝑥 (𝑛𝑥)) = 1 ∧ (𝑥 (𝑛𝑥)) = 0 )))))
49 df-oposet 33982 . . 3 OP = {𝑝 ∈ Poset ∣ (((Base‘𝑝) ∈ dom (lub‘𝑝) ∧ (Base‘𝑝) ∈ dom (glb‘𝑝)) ∧ ∃𝑛(𝑛 = (oc‘𝑝) ∧ ∀𝑥 ∈ (Base‘𝑝)∀𝑦 ∈ (Base‘𝑝)(((𝑛𝑥) ∈ (Base‘𝑝) ∧ (𝑛‘(𝑛𝑥)) = 𝑥 ∧ (𝑥(le‘𝑝)𝑦 → (𝑛𝑦)(le‘𝑝)(𝑛𝑥))) ∧ (𝑥(join‘𝑝)(𝑛𝑥)) = (1.‘𝑝) ∧ (𝑥(meet‘𝑝)(𝑛𝑥)) = (0.‘𝑝))))}
5048, 49elrab2 3353 . 2 (𝐾 ∈ OP ↔ (𝐾 ∈ Poset ∧ ((𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺) ∧ ∃𝑛(𝑛 = ∧ ∀𝑥𝐵𝑦𝐵 (((𝑛𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛𝑥)) = 𝑥 ∧ (𝑥 𝑦 → (𝑛𝑦) (𝑛𝑥))) ∧ (𝑥 (𝑛𝑥)) = 1 ∧ (𝑥 (𝑛𝑥)) = 0 )))))
51 anass 680 . 2 (((𝐾 ∈ Poset ∧ (𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺)) ∧ ∃𝑛(𝑛 = ∧ ∀𝑥𝐵𝑦𝐵 (((𝑛𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛𝑥)) = 𝑥 ∧ (𝑥 𝑦 → (𝑛𝑦) (𝑛𝑥))) ∧ (𝑥 (𝑛𝑥)) = 1 ∧ (𝑥 (𝑛𝑥)) = 0 ))) ↔ (𝐾 ∈ Poset ∧ ((𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺) ∧ ∃𝑛(𝑛 = ∧ ∀𝑥𝐵𝑦𝐵 (((𝑛𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛𝑥)) = 𝑥 ∧ (𝑥 𝑦 → (𝑛𝑦) (𝑛𝑥))) ∧ (𝑥 (𝑛𝑥)) = 1 ∧ (𝑥 (𝑛𝑥)) = 0 )))))
52 3anass 1040 . . . 4 ((𝐾 ∈ Poset ∧ 𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺) ↔ (𝐾 ∈ Poset ∧ (𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺)))
5352bicomi 214 . . 3 ((𝐾 ∈ Poset ∧ (𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺)) ↔ (𝐾 ∈ Poset ∧ 𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺))
54 fvex 6168 . . . . 5 (oc‘𝐾) ∈ V
5516, 54eqeltri 2694 . . . 4 ∈ V
56 fveq1 6157 . . . . . . . 8 (𝑛 = → (𝑛𝑥) = ( 𝑥))
5756eleq1d 2683 . . . . . . 7 (𝑛 = → ((𝑛𝑥) ∈ 𝐵 ↔ ( 𝑥) ∈ 𝐵))
58 id 22 . . . . . . . . 9 (𝑛 = 𝑛 = )
5958, 56fveq12d 6164 . . . . . . . 8 (𝑛 = → (𝑛‘(𝑛𝑥)) = ( ‘( 𝑥)))
6059eqeq1d 2623 . . . . . . 7 (𝑛 = → ((𝑛‘(𝑛𝑥)) = 𝑥 ↔ ( ‘( 𝑥)) = 𝑥))
61 fveq1 6157 . . . . . . . . 9 (𝑛 = → (𝑛𝑦) = ( 𝑦))
6261, 56breq12d 4636 . . . . . . . 8 (𝑛 = → ((𝑛𝑦) (𝑛𝑥) ↔ ( 𝑦) ( 𝑥)))
6362imbi2d 330 . . . . . . 7 (𝑛 = → ((𝑥 𝑦 → (𝑛𝑦) (𝑛𝑥)) ↔ (𝑥 𝑦 → ( 𝑦) ( 𝑥))))
6457, 60, 633anbi123d 1396 . . . . . 6 (𝑛 = → (((𝑛𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛𝑥)) = 𝑥 ∧ (𝑥 𝑦 → (𝑛𝑦) (𝑛𝑥))) ↔ (( 𝑥) ∈ 𝐵 ∧ ( ‘( 𝑥)) = 𝑥 ∧ (𝑥 𝑦 → ( 𝑦) ( 𝑥)))))
6556oveq2d 6631 . . . . . . 7 (𝑛 = → (𝑥 (𝑛𝑥)) = (𝑥 ( 𝑥)))
6665eqeq1d 2623 . . . . . 6 (𝑛 = → ((𝑥 (𝑛𝑥)) = 1 ↔ (𝑥 ( 𝑥)) = 1 ))
6756oveq2d 6631 . . . . . . 7 (𝑛 = → (𝑥 (𝑛𝑥)) = (𝑥 ( 𝑥)))
6867eqeq1d 2623 . . . . . 6 (𝑛 = → ((𝑥 (𝑛𝑥)) = 0 ↔ (𝑥 ( 𝑥)) = 0 ))
6964, 66, 683anbi123d 1396 . . . . 5 (𝑛 = → ((((𝑛𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛𝑥)) = 𝑥 ∧ (𝑥 𝑦 → (𝑛𝑦) (𝑛𝑥))) ∧ (𝑥 (𝑛𝑥)) = 1 ∧ (𝑥 (𝑛𝑥)) = 0 ) ↔ ((( 𝑥) ∈ 𝐵 ∧ ( ‘( 𝑥)) = 𝑥 ∧ (𝑥 𝑦 → ( 𝑦) ( 𝑥))) ∧ (𝑥 ( 𝑥)) = 1 ∧ (𝑥 ( 𝑥)) = 0 )))
70692ralbidv 2985 . . . 4 (𝑛 = → (∀𝑥𝐵𝑦𝐵 (((𝑛𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛𝑥)) = 𝑥 ∧ (𝑥 𝑦 → (𝑛𝑦) (𝑛𝑥))) ∧ (𝑥 (𝑛𝑥)) = 1 ∧ (𝑥 (𝑛𝑥)) = 0 ) ↔ ∀𝑥𝐵𝑦𝐵 ((( 𝑥) ∈ 𝐵 ∧ ( ‘( 𝑥)) = 𝑥 ∧ (𝑥 𝑦 → ( 𝑦) ( 𝑥))) ∧ (𝑥 ( 𝑥)) = 1 ∧ (𝑥 ( 𝑥)) = 0 )))
7155, 70ceqsexv 3232 . . 3 (∃𝑛(𝑛 = ∧ ∀𝑥𝐵𝑦𝐵 (((𝑛𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛𝑥)) = 𝑥 ∧ (𝑥 𝑦 → (𝑛𝑦) (𝑛𝑥))) ∧ (𝑥 (𝑛𝑥)) = 1 ∧ (𝑥 (𝑛𝑥)) = 0 )) ↔ ∀𝑥𝐵𝑦𝐵 ((( 𝑥) ∈ 𝐵 ∧ ( ‘( 𝑥)) = 𝑥 ∧ (𝑥 𝑦 → ( 𝑦) ( 𝑥))) ∧ (𝑥 ( 𝑥)) = 1 ∧ (𝑥 ( 𝑥)) = 0 ))
7253, 71anbi12i 732 . 2 (((𝐾 ∈ Poset ∧ (𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺)) ∧ ∃𝑛(𝑛 = ∧ ∀𝑥𝐵𝑦𝐵 (((𝑛𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛𝑥)) = 𝑥 ∧ (𝑥 𝑦 → (𝑛𝑦) (𝑛𝑥))) ∧ (𝑥 (𝑛𝑥)) = 1 ∧ (𝑥 (𝑛𝑥)) = 0 ))) ↔ ((𝐾 ∈ Poset ∧ 𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺) ∧ ∀𝑥𝐵𝑦𝐵 ((( 𝑥) ∈ 𝐵 ∧ ( ‘( 𝑥)) = 𝑥 ∧ (𝑥 𝑦 → ( 𝑦) ( 𝑥))) ∧ (𝑥 ( 𝑥)) = 1 ∧ (𝑥 ( 𝑥)) = 0 )))
7350, 51, 723bitr2i 288 1 (𝐾 ∈ OP ↔ ((𝐾 ∈ Poset ∧ 𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺) ∧ ∀𝑥𝐵𝑦𝐵 ((( 𝑥) ∈ 𝐵 ∧ ( ‘( 𝑥)) = 𝑥 ∧ (𝑥 𝑦 → ( 𝑦) ( 𝑥))) ∧ (𝑥 ( 𝑥)) = 1 ∧ (𝑥 ( 𝑥)) = 0 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wral 2908  Vcvv 3190   class class class wbr 4623  dom cdm 5084  cfv 5857  (class class class)co 6615  Basecbs 15800  lecple 15888  occoc 15889  Posetcpo 16880  lubclub 16882  glbcglb 16883  joincjn 16884  meetcmee 16885  0.cp0 16977  1.cp1 16978  OPcops 33978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4759
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-dm 5094  df-iota 5820  df-fv 5865  df-ov 6618  df-oposet 33982
This theorem is referenced by:  opposet  33987  oposlem  33988  op01dm  33989
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