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Theorem isopos 36186
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 6666 . . . . . . 7 (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
2 isopos.b . . . . . . 7 𝐵 = (Base‘𝐾)
31, 2syl6eqr 2878 . . . . . 6 (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
4 fveq2 6666 . . . . . . . 8 (𝑝 = 𝐾 → (lub‘𝑝) = (lub‘𝐾))
5 isopos.e . . . . . . . 8 𝑈 = (lub‘𝐾)
64, 5syl6eqr 2878 . . . . . . 7 (𝑝 = 𝐾 → (lub‘𝑝) = 𝑈)
76dmeqd 5772 . . . . . 6 (𝑝 = 𝐾 → dom (lub‘𝑝) = dom 𝑈)
83, 7eleq12d 2911 . . . . 5 (𝑝 = 𝐾 → ((Base‘𝑝) ∈ dom (lub‘𝑝) ↔ 𝐵 ∈ dom 𝑈))
9 fveq2 6666 . . . . . . . 8 (𝑝 = 𝐾 → (glb‘𝑝) = (glb‘𝐾))
10 isopos.g . . . . . . . 8 𝐺 = (glb‘𝐾)
119, 10syl6eqr 2878 . . . . . . 7 (𝑝 = 𝐾 → (glb‘𝑝) = 𝐺)
1211dmeqd 5772 . . . . . 6 (𝑝 = 𝐾 → dom (glb‘𝑝) = dom 𝐺)
133, 12eleq12d 2911 . . . . 5 (𝑝 = 𝐾 → ((Base‘𝑝) ∈ dom (glb‘𝑝) ↔ 𝐵 ∈ dom 𝐺))
148, 13anbi12d 630 . . . 4 (𝑝 = 𝐾 → (((Base‘𝑝) ∈ dom (lub‘𝑝) ∧ (Base‘𝑝) ∈ dom (glb‘𝑝)) ↔ (𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺)))
15 fveq2 6666 . . . . . . . 8 (𝑝 = 𝐾 → (oc‘𝑝) = (oc‘𝐾))
16 isopos.o . . . . . . . 8 = (oc‘𝐾)
1715, 16syl6eqr 2878 . . . . . . 7 (𝑝 = 𝐾 → (oc‘𝑝) = )
1817eqeq2d 2836 . . . . . 6 (𝑝 = 𝐾 → (𝑛 = (oc‘𝑝) ↔ 𝑛 = ))
193eleq2d 2902 . . . . . . . . . 10 (𝑝 = 𝐾 → ((𝑛𝑥) ∈ (Base‘𝑝) ↔ (𝑛𝑥) ∈ 𝐵))
20 fveq2 6666 . . . . . . . . . . . . 13 (𝑝 = 𝐾 → (le‘𝑝) = (le‘𝐾))
21 isopos.l . . . . . . . . . . . . 13 = (le‘𝐾)
2220, 21syl6eqr 2878 . . . . . . . . . . . 12 (𝑝 = 𝐾 → (le‘𝑝) = )
2322breqd 5073 . . . . . . . . . . 11 (𝑝 = 𝐾 → (𝑥(le‘𝑝)𝑦𝑥 𝑦))
2422breqd 5073 . . . . . . . . . . 11 (𝑝 = 𝐾 → ((𝑛𝑦)(le‘𝑝)(𝑛𝑥) ↔ (𝑛𝑦) (𝑛𝑥)))
2523, 24imbi12d 346 . . . . . . . . . 10 (𝑝 = 𝐾 → ((𝑥(le‘𝑝)𝑦 → (𝑛𝑦)(le‘𝑝)(𝑛𝑥)) ↔ (𝑥 𝑦 → (𝑛𝑦) (𝑛𝑥))))
2619, 253anbi13d 1431 . . . . . . . . 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 6666 . . . . . . . . . . . 12 (𝑝 = 𝐾 → (join‘𝑝) = (join‘𝐾))
28 isopos.j . . . . . . . . . . . 12 = (join‘𝐾)
2927, 28syl6eqr 2878 . . . . . . . . . . 11 (𝑝 = 𝐾 → (join‘𝑝) = )
3029oveqd 7168 . . . . . . . . . 10 (𝑝 = 𝐾 → (𝑥(join‘𝑝)(𝑛𝑥)) = (𝑥 (𝑛𝑥)))
31 fveq2 6666 . . . . . . . . . . 11 (𝑝 = 𝐾 → (1.‘𝑝) = (1.‘𝐾))
32 isopos.u . . . . . . . . . . 11 1 = (1.‘𝐾)
3331, 32syl6eqr 2878 . . . . . . . . . 10 (𝑝 = 𝐾 → (1.‘𝑝) = 1 )
3430, 33eqeq12d 2841 . . . . . . . . 9 (𝑝 = 𝐾 → ((𝑥(join‘𝑝)(𝑛𝑥)) = (1.‘𝑝) ↔ (𝑥 (𝑛𝑥)) = 1 ))
35 fveq2 6666 . . . . . . . . . . . 12 (𝑝 = 𝐾 → (meet‘𝑝) = (meet‘𝐾))
36 isopos.m . . . . . . . . . . . 12 = (meet‘𝐾)
3735, 36syl6eqr 2878 . . . . . . . . . . 11 (𝑝 = 𝐾 → (meet‘𝑝) = )
3837oveqd 7168 . . . . . . . . . 10 (𝑝 = 𝐾 → (𝑥(meet‘𝑝)(𝑛𝑥)) = (𝑥 (𝑛𝑥)))
39 fveq2 6666 . . . . . . . . . . 11 (𝑝 = 𝐾 → (0.‘𝑝) = (0.‘𝐾))
40 isopos.f . . . . . . . . . . 11 0 = (0.‘𝐾)
4139, 40syl6eqr 2878 . . . . . . . . . 10 (𝑝 = 𝐾 → (0.‘𝑝) = 0 )
4238, 41eqeq12d 2841 . . . . . . . . 9 (𝑝 = 𝐾 → ((𝑥(meet‘𝑝)(𝑛𝑥)) = (0.‘𝑝) ↔ (𝑥 (𝑛𝑥)) = 0 ))
4326, 34, 423anbi123d 1429 . . . . . . . 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 3406 . . . . . . 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 3406 . . . . . 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 630 . . . . 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 1915 . . . 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 630 . . 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 36182 . . 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 3686 . 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 469 . 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 1089 . . . 4 ((𝐾 ∈ Poset ∧ 𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺) ↔ (𝐾 ∈ Poset ∧ (𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺)))
5352bicomi 225 . . 3 ((𝐾 ∈ Poset ∧ (𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺)) ↔ (𝐾 ∈ Poset ∧ 𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺))
5416fvexi 6680 . . . 4 ∈ V
55 fveq1 6665 . . . . . . . 8 (𝑛 = → (𝑛𝑥) = ( 𝑥))
5655eleq1d 2901 . . . . . . 7 (𝑛 = → ((𝑛𝑥) ∈ 𝐵 ↔ ( 𝑥) ∈ 𝐵))
57 id 22 . . . . . . . . 9 (𝑛 = 𝑛 = )
5857, 55fveq12d 6673 . . . . . . . 8 (𝑛 = → (𝑛‘(𝑛𝑥)) = ( ‘( 𝑥)))
5958eqeq1d 2827 . . . . . . 7 (𝑛 = → ((𝑛‘(𝑛𝑥)) = 𝑥 ↔ ( ‘( 𝑥)) = 𝑥))
60 fveq1 6665 . . . . . . . . 9 (𝑛 = → (𝑛𝑦) = ( 𝑦))
6160, 55breq12d 5075 . . . . . . . 8 (𝑛 = → ((𝑛𝑦) (𝑛𝑥) ↔ ( 𝑦) ( 𝑥)))
6261imbi2d 342 . . . . . . 7 (𝑛 = → ((𝑥 𝑦 → (𝑛𝑦) (𝑛𝑥)) ↔ (𝑥 𝑦 → ( 𝑦) ( 𝑥))))
6356, 59, 623anbi123d 1429 . . . . . 6 (𝑛 = → (((𝑛𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛𝑥)) = 𝑥 ∧ (𝑥 𝑦 → (𝑛𝑦) (𝑛𝑥))) ↔ (( 𝑥) ∈ 𝐵 ∧ ( ‘( 𝑥)) = 𝑥 ∧ (𝑥 𝑦 → ( 𝑦) ( 𝑥)))))
6455oveq2d 7167 . . . . . . 7 (𝑛 = → (𝑥 (𝑛𝑥)) = (𝑥 ( 𝑥)))
6564eqeq1d 2827 . . . . . 6 (𝑛 = → ((𝑥 (𝑛𝑥)) = 1 ↔ (𝑥 ( 𝑥)) = 1 ))
6655oveq2d 7167 . . . . . . 7 (𝑛 = → (𝑥 (𝑛𝑥)) = (𝑥 ( 𝑥)))
6766eqeq1d 2827 . . . . . 6 (𝑛 = → ((𝑥 (𝑛𝑥)) = 0 ↔ (𝑥 ( 𝑥)) = 0 ))
6863, 65, 673anbi123d 1429 . . . . 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 3203 . . . 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 3546 . . 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 626 . 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 300 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 207  wa 396  w3a 1081   = wceq 1530  wex 1773  wcel 2107  wral 3142   class class class wbr 5062  dom cdm 5553  cfv 6351  (class class class)co 7151  Basecbs 16476  lecple 16565  occoc 16566  Posetcpo 17543  lubclub 17545  glbcglb 17546  joincjn 17547  meetcmee 17548  0.cp0 17640  1.cp1 17641  OPcops 36178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-nul 5206
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-dm 5563  df-iota 6311  df-fv 6359  df-ov 7154  df-oposet 36182
This theorem is referenced by:  opposet  36187  oposlem  36188  op01dm  36189
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