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Theorem isopos 39181
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 6906 . . . . . . 7 (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
2 isopos.b . . . . . . 7 𝐵 = (Base‘𝐾)
31, 2eqtr4di 2795 . . . . . 6 (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
4 fveq2 6906 . . . . . . . 8 (𝑝 = 𝐾 → (lub‘𝑝) = (lub‘𝐾))
5 isopos.e . . . . . . . 8 𝑈 = (lub‘𝐾)
64, 5eqtr4di 2795 . . . . . . 7 (𝑝 = 𝐾 → (lub‘𝑝) = 𝑈)
76dmeqd 5916 . . . . . 6 (𝑝 = 𝐾 → dom (lub‘𝑝) = dom 𝑈)
83, 7eleq12d 2835 . . . . 5 (𝑝 = 𝐾 → ((Base‘𝑝) ∈ dom (lub‘𝑝) ↔ 𝐵 ∈ dom 𝑈))
9 fveq2 6906 . . . . . . . 8 (𝑝 = 𝐾 → (glb‘𝑝) = (glb‘𝐾))
10 isopos.g . . . . . . . 8 𝐺 = (glb‘𝐾)
119, 10eqtr4di 2795 . . . . . . 7 (𝑝 = 𝐾 → (glb‘𝑝) = 𝐺)
1211dmeqd 5916 . . . . . 6 (𝑝 = 𝐾 → dom (glb‘𝑝) = dom 𝐺)
133, 12eleq12d 2835 . . . . 5 (𝑝 = 𝐾 → ((Base‘𝑝) ∈ dom (glb‘𝑝) ↔ 𝐵 ∈ dom 𝐺))
148, 13anbi12d 632 . . . 4 (𝑝 = 𝐾 → (((Base‘𝑝) ∈ dom (lub‘𝑝) ∧ (Base‘𝑝) ∈ dom (glb‘𝑝)) ↔ (𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺)))
15 fveq2 6906 . . . . . . . 8 (𝑝 = 𝐾 → (oc‘𝑝) = (oc‘𝐾))
16 isopos.o . . . . . . . 8 = (oc‘𝐾)
1715, 16eqtr4di 2795 . . . . . . 7 (𝑝 = 𝐾 → (oc‘𝑝) = )
1817eqeq2d 2748 . . . . . 6 (𝑝 = 𝐾 → (𝑛 = (oc‘𝑝) ↔ 𝑛 = ))
193eleq2d 2827 . . . . . . . . . 10 (𝑝 = 𝐾 → ((𝑛𝑥) ∈ (Base‘𝑝) ↔ (𝑛𝑥) ∈ 𝐵))
20 fveq2 6906 . . . . . . . . . . . . 13 (𝑝 = 𝐾 → (le‘𝑝) = (le‘𝐾))
21 isopos.l . . . . . . . . . . . . 13 = (le‘𝐾)
2220, 21eqtr4di 2795 . . . . . . . . . . . 12 (𝑝 = 𝐾 → (le‘𝑝) = )
2322breqd 5154 . . . . . . . . . . 11 (𝑝 = 𝐾 → (𝑥(le‘𝑝)𝑦𝑥 𝑦))
2422breqd 5154 . . . . . . . . . . 11 (𝑝 = 𝐾 → ((𝑛𝑦)(le‘𝑝)(𝑛𝑥) ↔ (𝑛𝑦) (𝑛𝑥)))
2523, 24imbi12d 344 . . . . . . . . . 10 (𝑝 = 𝐾 → ((𝑥(le‘𝑝)𝑦 → (𝑛𝑦)(le‘𝑝)(𝑛𝑥)) ↔ (𝑥 𝑦 → (𝑛𝑦) (𝑛𝑥))))
2619, 253anbi13d 1440 . . . . . . . . 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 6906 . . . . . . . . . . . 12 (𝑝 = 𝐾 → (join‘𝑝) = (join‘𝐾))
28 isopos.j . . . . . . . . . . . 12 = (join‘𝐾)
2927, 28eqtr4di 2795 . . . . . . . . . . 11 (𝑝 = 𝐾 → (join‘𝑝) = )
3029oveqd 7448 . . . . . . . . . 10 (𝑝 = 𝐾 → (𝑥(join‘𝑝)(𝑛𝑥)) = (𝑥 (𝑛𝑥)))
31 fveq2 6906 . . . . . . . . . . 11 (𝑝 = 𝐾 → (1.‘𝑝) = (1.‘𝐾))
32 isopos.u . . . . . . . . . . 11 1 = (1.‘𝐾)
3331, 32eqtr4di 2795 . . . . . . . . . 10 (𝑝 = 𝐾 → (1.‘𝑝) = 1 )
3430, 33eqeq12d 2753 . . . . . . . . 9 (𝑝 = 𝐾 → ((𝑥(join‘𝑝)(𝑛𝑥)) = (1.‘𝑝) ↔ (𝑥 (𝑛𝑥)) = 1 ))
35 fveq2 6906 . . . . . . . . . . . 12 (𝑝 = 𝐾 → (meet‘𝑝) = (meet‘𝐾))
36 isopos.m . . . . . . . . . . . 12 = (meet‘𝐾)
3735, 36eqtr4di 2795 . . . . . . . . . . 11 (𝑝 = 𝐾 → (meet‘𝑝) = )
3837oveqd 7448 . . . . . . . . . 10 (𝑝 = 𝐾 → (𝑥(meet‘𝑝)(𝑛𝑥)) = (𝑥 (𝑛𝑥)))
39 fveq2 6906 . . . . . . . . . . 11 (𝑝 = 𝐾 → (0.‘𝑝) = (0.‘𝐾))
40 isopos.f . . . . . . . . . . 11 0 = (0.‘𝐾)
4139, 40eqtr4di 2795 . . . . . . . . . 10 (𝑝 = 𝐾 → (0.‘𝑝) = 0 )
4238, 41eqeq12d 2753 . . . . . . . . 9 (𝑝 = 𝐾 → ((𝑥(meet‘𝑝)(𝑛𝑥)) = (0.‘𝑝) ↔ (𝑥 (𝑛𝑥)) = 0 ))
4326, 34, 423anbi123d 1438 . . . . . . . 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 3346 . . . . . . 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 3346 . . . . . 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 632 . . . . 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 1921 . . . 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 632 . . 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 39177 . . 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 3695 . 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 468 . 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 1095 . . . 4 ((𝐾 ∈ Poset ∧ 𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺) ↔ (𝐾 ∈ Poset ∧ (𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺)))
5352bicomi 224 . . 3 ((𝐾 ∈ Poset ∧ (𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺)) ↔ (𝐾 ∈ Poset ∧ 𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺))
5416fvexi 6920 . . . 4 ∈ V
55 fveq1 6905 . . . . . . . 8 (𝑛 = → (𝑛𝑥) = ( 𝑥))
5655eleq1d 2826 . . . . . . 7 (𝑛 = → ((𝑛𝑥) ∈ 𝐵 ↔ ( 𝑥) ∈ 𝐵))
57 id 22 . . . . . . . . 9 (𝑛 = 𝑛 = )
5857, 55fveq12d 6913 . . . . . . . 8 (𝑛 = → (𝑛‘(𝑛𝑥)) = ( ‘( 𝑥)))
5958eqeq1d 2739 . . . . . . 7 (𝑛 = → ((𝑛‘(𝑛𝑥)) = 𝑥 ↔ ( ‘( 𝑥)) = 𝑥))
60 fveq1 6905 . . . . . . . . 9 (𝑛 = → (𝑛𝑦) = ( 𝑦))
6160, 55breq12d 5156 . . . . . . . 8 (𝑛 = → ((𝑛𝑦) (𝑛𝑥) ↔ ( 𝑦) ( 𝑥)))
6261imbi2d 340 . . . . . . 7 (𝑛 = → ((𝑥 𝑦 → (𝑛𝑦) (𝑛𝑥)) ↔ (𝑥 𝑦 → ( 𝑦) ( 𝑥))))
6356, 59, 623anbi123d 1438 . . . . . 6 (𝑛 = → (((𝑛𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛𝑥)) = 𝑥 ∧ (𝑥 𝑦 → (𝑛𝑦) (𝑛𝑥))) ↔ (( 𝑥) ∈ 𝐵 ∧ ( ‘( 𝑥)) = 𝑥 ∧ (𝑥 𝑦 → ( 𝑦) ( 𝑥)))))
6455oveq2d 7447 . . . . . . 7 (𝑛 = → (𝑥 (𝑛𝑥)) = (𝑥 ( 𝑥)))
6564eqeq1d 2739 . . . . . 6 (𝑛 = → ((𝑥 (𝑛𝑥)) = 1 ↔ (𝑥 ( 𝑥)) = 1 ))
6655oveq2d 7447 . . . . . . 7 (𝑛 = → (𝑥 (𝑛𝑥)) = (𝑥 ( 𝑥)))
6766eqeq1d 2739 . . . . . 6 (𝑛 = → ((𝑥 (𝑛𝑥)) = 0 ↔ (𝑥 ( 𝑥)) = 0 ))
6863, 65, 673anbi123d 1438 . . . . 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 )))
69682ralbidv 3221 . . . 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 )))
7054, 69ceqsexv 3532 . . 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 ))
7153, 70anbi12i 628 . 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 )))
7250, 51, 713bitr2i 299 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 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wral 3061   class class class wbr 5143  dom cdm 5685  cfv 6561  (class class class)co 7431  Basecbs 17247  lecple 17304  occoc 17305  Posetcpo 18353  lubclub 18355  glbcglb 18356  joincjn 18357  meetcmee 18358  0.cp0 18468  1.cp1 18469  OPcops 39173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-dm 5695  df-iota 6514  df-fv 6569  df-ov 7434  df-oposet 39177
This theorem is referenced by:  opposet  39182  oposlem  39183  op01dm  39184
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