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Theorem isopos 34785
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 6229 . . . . . . 7 (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
2 isopos.b . . . . . . 7 𝐵 = (Base‘𝐾)
31, 2syl6eqr 2703 . . . . . 6 (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
4 fveq2 6229 . . . . . . . 8 (𝑝 = 𝐾 → (lub‘𝑝) = (lub‘𝐾))
5 isopos.e . . . . . . . 8 𝑈 = (lub‘𝐾)
64, 5syl6eqr 2703 . . . . . . 7 (𝑝 = 𝐾 → (lub‘𝑝) = 𝑈)
76dmeqd 5358 . . . . . 6 (𝑝 = 𝐾 → dom (lub‘𝑝) = dom 𝑈)
83, 7eleq12d 2724 . . . . 5 (𝑝 = 𝐾 → ((Base‘𝑝) ∈ dom (lub‘𝑝) ↔ 𝐵 ∈ dom 𝑈))
9 fveq2 6229 . . . . . . . 8 (𝑝 = 𝐾 → (glb‘𝑝) = (glb‘𝐾))
10 isopos.g . . . . . . . 8 𝐺 = (glb‘𝐾)
119, 10syl6eqr 2703 . . . . . . 7 (𝑝 = 𝐾 → (glb‘𝑝) = 𝐺)
1211dmeqd 5358 . . . . . 6 (𝑝 = 𝐾 → dom (glb‘𝑝) = dom 𝐺)
133, 12eleq12d 2724 . . . . 5 (𝑝 = 𝐾 → ((Base‘𝑝) ∈ dom (glb‘𝑝) ↔ 𝐵 ∈ dom 𝐺))
148, 13anbi12d 747 . . . 4 (𝑝 = 𝐾 → (((Base‘𝑝) ∈ dom (lub‘𝑝) ∧ (Base‘𝑝) ∈ dom (glb‘𝑝)) ↔ (𝐵 ∈ dom 𝑈𝐵 ∈ dom 𝐺)))
15 fveq2 6229 . . . . . . . 8 (𝑝 = 𝐾 → (oc‘𝑝) = (oc‘𝐾))
16 isopos.o . . . . . . . 8 = (oc‘𝐾)
1715, 16syl6eqr 2703 . . . . . . 7 (𝑝 = 𝐾 → (oc‘𝑝) = )
1817eqeq2d 2661 . . . . . 6 (𝑝 = 𝐾 → (𝑛 = (oc‘𝑝) ↔ 𝑛 = ))
193eleq2d 2716 . . . . . . . . . 10 (𝑝 = 𝐾 → ((𝑛𝑥) ∈ (Base‘𝑝) ↔ (𝑛𝑥) ∈ 𝐵))
20 fveq2 6229 . . . . . . . . . . . . 13 (𝑝 = 𝐾 → (le‘𝑝) = (le‘𝐾))
21 isopos.l . . . . . . . . . . . . 13 = (le‘𝐾)
2220, 21syl6eqr 2703 . . . . . . . . . . . 12 (𝑝 = 𝐾 → (le‘𝑝) = )
2322breqd 4696 . . . . . . . . . . 11 (𝑝 = 𝐾 → (𝑥(le‘𝑝)𝑦𝑥 𝑦))
2422breqd 4696 . . . . . . . . . . 11 (𝑝 = 𝐾 → ((𝑛𝑦)(le‘𝑝)(𝑛𝑥) ↔ (𝑛𝑦) (𝑛𝑥)))
2523, 24imbi12d 333 . . . . . . . . . 10 (𝑝 = 𝐾 → ((𝑥(le‘𝑝)𝑦 → (𝑛𝑦)(le‘𝑝)(𝑛𝑥)) ↔ (𝑥 𝑦 → (𝑛𝑦) (𝑛𝑥))))
2619, 253anbi13d 1441 . . . . . . . . 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 6229 . . . . . . . . . . . 12 (𝑝 = 𝐾 → (join‘𝑝) = (join‘𝐾))
28 isopos.j . . . . . . . . . . . 12 = (join‘𝐾)
2927, 28syl6eqr 2703 . . . . . . . . . . 11 (𝑝 = 𝐾 → (join‘𝑝) = )
3029oveqd 6707 . . . . . . . . . 10 (𝑝 = 𝐾 → (𝑥(join‘𝑝)(𝑛𝑥)) = (𝑥 (𝑛𝑥)))
31 fveq2 6229 . . . . . . . . . . 11 (𝑝 = 𝐾 → (1.‘𝑝) = (1.‘𝐾))
32 isopos.u . . . . . . . . . . 11 1 = (1.‘𝐾)
3331, 32syl6eqr 2703 . . . . . . . . . 10 (𝑝 = 𝐾 → (1.‘𝑝) = 1 )
3430, 33eqeq12d 2666 . . . . . . . . 9 (𝑝 = 𝐾 → ((𝑥(join‘𝑝)(𝑛𝑥)) = (1.‘𝑝) ↔ (𝑥 (𝑛𝑥)) = 1 ))
35 fveq2 6229 . . . . . . . . . . . 12 (𝑝 = 𝐾 → (meet‘𝑝) = (meet‘𝐾))
36 isopos.m . . . . . . . . . . . 12 = (meet‘𝐾)
3735, 36syl6eqr 2703 . . . . . . . . . . 11 (𝑝 = 𝐾 → (meet‘𝑝) = )
3837oveqd 6707 . . . . . . . . . 10 (𝑝 = 𝐾 → (𝑥(meet‘𝑝)(𝑛𝑥)) = (𝑥 (𝑛𝑥)))
39 fveq2 6229 . . . . . . . . . . 11 (𝑝 = 𝐾 → (0.‘𝑝) = (0.‘𝐾))
40 isopos.f . . . . . . . . . . 11 0 = (0.‘𝐾)
4139, 40syl6eqr 2703 . . . . . . . . . 10 (𝑝 = 𝐾 → (0.‘𝑝) = 0 )
4238, 41eqeq12d 2666 . . . . . . . . 9 (𝑝 = 𝐾 → ((𝑥(meet‘𝑝)(𝑛𝑥)) = (0.‘𝑝) ↔ (𝑥 (𝑛𝑥)) = 0 ))
4326, 34, 423anbi123d 1439 . . . . . . . 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 3182 . . . . . . 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 3182 . . . . . 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 747 . . . . 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 1890 . . . 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 747 . . 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 34781 . . 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 3399 . 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 682 . 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 1059 . . . 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 6239 . . . . 5 (oc‘𝐾) ∈ V
5516, 54eqeltri 2726 . . . 4 ∈ V
56 fveq1 6228 . . . . . . . 8 (𝑛 = → (𝑛𝑥) = ( 𝑥))
5756eleq1d 2715 . . . . . . 7 (𝑛 = → ((𝑛𝑥) ∈ 𝐵 ↔ ( 𝑥) ∈ 𝐵))
58 id 22 . . . . . . . . 9 (𝑛 = 𝑛 = )
5958, 56fveq12d 6235 . . . . . . . 8 (𝑛 = → (𝑛‘(𝑛𝑥)) = ( ‘( 𝑥)))
6059eqeq1d 2653 . . . . . . 7 (𝑛 = → ((𝑛‘(𝑛𝑥)) = 𝑥 ↔ ( ‘( 𝑥)) = 𝑥))
61 fveq1 6228 . . . . . . . . 9 (𝑛 = → (𝑛𝑦) = ( 𝑦))
6261, 56breq12d 4698 . . . . . . . 8 (𝑛 = → ((𝑛𝑦) (𝑛𝑥) ↔ ( 𝑦) ( 𝑥)))
6362imbi2d 329 . . . . . . 7 (𝑛 = → ((𝑥 𝑦 → (𝑛𝑦) (𝑛𝑥)) ↔ (𝑥 𝑦 → ( 𝑦) ( 𝑥))))
6457, 60, 633anbi123d 1439 . . . . . 6 (𝑛 = → (((𝑛𝑥) ∈ 𝐵 ∧ (𝑛‘(𝑛𝑥)) = 𝑥 ∧ (𝑥 𝑦 → (𝑛𝑦) (𝑛𝑥))) ↔ (( 𝑥) ∈ 𝐵 ∧ ( ‘( 𝑥)) = 𝑥 ∧ (𝑥 𝑦 → ( 𝑦) ( 𝑥)))))
6556oveq2d 6706 . . . . . . 7 (𝑛 = → (𝑥 (𝑛𝑥)) = (𝑥 ( 𝑥)))
6665eqeq1d 2653 . . . . . 6 (𝑛 = → ((𝑥 (𝑛𝑥)) = 1 ↔ (𝑥 ( 𝑥)) = 1 ))
6756oveq2d 6706 . . . . . . 7 (𝑛 = → (𝑥 (𝑛𝑥)) = (𝑥 ( 𝑥)))
6867eqeq1d 2653 . . . . . 6 (𝑛 = → ((𝑥 (𝑛𝑥)) = 0 ↔ (𝑥 ( 𝑥)) = 0 ))
6964, 66, 683anbi123d 1439 . . . . 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 3018 . . . 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 3273 . . 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 733 . 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 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  wral 2941  Vcvv 3231   class class class wbr 4685  dom cdm 5143  cfv 5926  (class class class)co 6690  Basecbs 15904  lecple 15995  occoc 15996  Posetcpo 16987  lubclub 16989  glbcglb 16990  joincjn 16991  meetcmee 16992  0.cp0 17084  1.cp1 17085  OPcops 34777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-dm 5153  df-iota 5889  df-fv 5934  df-ov 6693  df-oposet 34781
This theorem is referenced by:  opposet  34786  oposlem  34787  op01dm  34788
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