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Theorem finomni 7112
Description: A finite set is omniscient. Remark right after Definition 3.1 of [Pierik], p. 14. (Contributed by Jim Kingdon, 28-Jun-2022.)
Assertion
Ref Expression
finomni (𝐴 ∈ Fin → 𝐴 ∈ Omni)

Proof of Theorem finomni
Dummy variables 𝑤 𝑦 𝑧 𝑓 𝑔 𝑥 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2233 . 2 (𝑤 = ∅ → (𝑤 ∈ Omni ↔ ∅ ∈ Omni))
2 eleq1 2233 . 2 (𝑤 = 𝑦 → (𝑤 ∈ Omni ↔ 𝑦 ∈ Omni))
3 eleq1 2233 . 2 (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 ∈ Omni ↔ (𝑦 ∪ {𝑧}) ∈ Omni))
4 eleq1 2233 . 2 (𝑤 = 𝐴 → (𝑤 ∈ Omni ↔ 𝐴 ∈ Omni))
5 0ex 4114 . . . 4 ∅ ∈ V
6 isomni 7108 . . . 4 (∅ ∈ V → (∅ ∈ Omni ↔ ∀𝑓(𝑓:∅⟶2o → (∃𝑥 ∈ ∅ (𝑓𝑥) = ∅ ∨ ∀𝑥 ∈ ∅ (𝑓𝑥) = 1o))))
75, 6ax-mp 5 . . 3 (∅ ∈ Omni ↔ ∀𝑓(𝑓:∅⟶2o → (∃𝑥 ∈ ∅ (𝑓𝑥) = ∅ ∨ ∀𝑥 ∈ ∅ (𝑓𝑥) = 1o)))
8 ral0 3515 . . . . 5 𝑥 ∈ ∅ (𝑓𝑥) = 1o
98olci 727 . . . 4 (∃𝑥 ∈ ∅ (𝑓𝑥) = ∅ ∨ ∀𝑥 ∈ ∅ (𝑓𝑥) = 1o)
109a1i 9 . . 3 (𝑓:∅⟶2o → (∃𝑥 ∈ ∅ (𝑓𝑥) = ∅ ∨ ∀𝑥 ∈ ∅ (𝑓𝑥) = 1o))
117, 10mpgbir 1446 . 2 ∅ ∈ Omni
12 elun1 3294 . . . . . . . . . . . 12 (𝑥𝑦𝑥 ∈ (𝑦 ∪ {𝑧}))
1312ad2antlr 486 . . . . . . . . . . 11 (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ 𝑥𝑦) ∧ ((𝑔𝑦)‘𝑥) = ∅) → 𝑥 ∈ (𝑦 ∪ {𝑧}))
14 fvres 5518 . . . . . . . . . . . . 13 (𝑥𝑦 → ((𝑔𝑦)‘𝑥) = (𝑔𝑥))
1514ad2antlr 486 . . . . . . . . . . . 12 (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ 𝑥𝑦) ∧ ((𝑔𝑦)‘𝑥) = ∅) → ((𝑔𝑦)‘𝑥) = (𝑔𝑥))
16 simpr 109 . . . . . . . . . . . 12 (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ 𝑥𝑦) ∧ ((𝑔𝑦)‘𝑥) = ∅) → ((𝑔𝑦)‘𝑥) = ∅)
1715, 16eqtr3d 2205 . . . . . . . . . . 11 (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ 𝑥𝑦) ∧ ((𝑔𝑦)‘𝑥) = ∅) → (𝑔𝑥) = ∅)
18 fveq2 5494 . . . . . . . . . . . . 13 (𝑢 = 𝑥 → (𝑔𝑢) = (𝑔𝑥))
1918eqeq1d 2179 . . . . . . . . . . . 12 (𝑢 = 𝑥 → ((𝑔𝑢) = ∅ ↔ (𝑔𝑥) = ∅))
2019rspcev 2834 . . . . . . . . . . 11 ((𝑥 ∈ (𝑦 ∪ {𝑧}) ∧ (𝑔𝑥) = ∅) → ∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅)
2113, 17, 20syl2anc 409 . . . . . . . . . 10 (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ 𝑥𝑦) ∧ ((𝑔𝑦)‘𝑥) = ∅) → ∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅)
2221orcd 728 . . . . . . . . 9 (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ 𝑥𝑦) ∧ ((𝑔𝑦)‘𝑥) = ∅) → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1o))
2322ex 114 . . . . . . . 8 ((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ 𝑥𝑦) → (((𝑔𝑦)‘𝑥) = ∅ → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1o)))
2423rexlimdva 2587 . . . . . . 7 (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) → (∃𝑥𝑦 ((𝑔𝑦)‘𝑥) = ∅ → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1o)))
25 vsnid 3613 . . . . . . . . . . . . 13 𝑧 ∈ {𝑧}
26 elun2 3295 . . . . . . . . . . . . 13 (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝑦 ∪ {𝑧}))
2725, 26ax-mp 5 . . . . . . . . . . . 12 𝑧 ∈ (𝑦 ∪ {𝑧})
2827a1i 9 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) → 𝑧 ∈ (𝑦 ∪ {𝑧}))
29 fveq2 5494 . . . . . . . . . . . . 13 (𝑢 = 𝑧 → (𝑔𝑢) = (𝑔𝑧))
3029eqeq1d 2179 . . . . . . . . . . . 12 (𝑢 = 𝑧 → ((𝑔𝑢) = ∅ ↔ (𝑔𝑧) = ∅))
3130rspcev 2834 . . . . . . . . . . 11 ((𝑧 ∈ (𝑦 ∪ {𝑧}) ∧ (𝑔𝑧) = ∅) → ∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅)
3228, 31sylan 281 . . . . . . . . . 10 ((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ (𝑔𝑧) = ∅) → ∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅)
3332orcd 728 . . . . . . . . 9 ((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ (𝑔𝑧) = ∅) → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1o))
3433a1d 22 . . . . . . . 8 ((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ (𝑔𝑧) = ∅) → (∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1o → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1o)))
35 simpr 109 . . . . . . . . . . . 12 (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ (𝑔𝑧) = 1o) ∧ ∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1o) → ∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1o)
36 fveq2 5494 . . . . . . . . . . . . . . 15 (𝑥 = 𝑢 → ((𝑔𝑦)‘𝑥) = ((𝑔𝑦)‘𝑢))
3736eqeq1d 2179 . . . . . . . . . . . . . 14 (𝑥 = 𝑢 → (((𝑔𝑦)‘𝑥) = 1o ↔ ((𝑔𝑦)‘𝑢) = 1o))
3837cbvralv 2696 . . . . . . . . . . . . 13 (∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1o ↔ ∀𝑢𝑦 ((𝑔𝑦)‘𝑢) = 1o)
39 fvres 5518 . . . . . . . . . . . . . . 15 (𝑢𝑦 → ((𝑔𝑦)‘𝑢) = (𝑔𝑢))
4039eqeq1d 2179 . . . . . . . . . . . . . 14 (𝑢𝑦 → (((𝑔𝑦)‘𝑢) = 1o ↔ (𝑔𝑢) = 1o))
4140ralbiia 2484 . . . . . . . . . . . . 13 (∀𝑢𝑦 ((𝑔𝑦)‘𝑢) = 1o ↔ ∀𝑢𝑦 (𝑔𝑢) = 1o)
4238, 41bitri 183 . . . . . . . . . . . 12 (∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1o ↔ ∀𝑢𝑦 (𝑔𝑢) = 1o)
4335, 42sylib 121 . . . . . . . . . . 11 (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ (𝑔𝑧) = 1o) ∧ ∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1o) → ∀𝑢𝑦 (𝑔𝑢) = 1o)
44 simplr 525 . . . . . . . . . . . 12 (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ (𝑔𝑧) = 1o) ∧ ∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1o) → (𝑔𝑧) = 1o)
45 vex 2733 . . . . . . . . . . . . 13 𝑧 ∈ V
4629eqeq1d 2179 . . . . . . . . . . . . 13 (𝑢 = 𝑧 → ((𝑔𝑢) = 1o ↔ (𝑔𝑧) = 1o))
4745, 46ralsn 3624 . . . . . . . . . . . 12 (∀𝑢 ∈ {𝑧} (𝑔𝑢) = 1o ↔ (𝑔𝑧) = 1o)
4844, 47sylibr 133 . . . . . . . . . . 11 (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ (𝑔𝑧) = 1o) ∧ ∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1o) → ∀𝑢 ∈ {𝑧} (𝑔𝑢) = 1o)
49 ralun 3309 . . . . . . . . . . 11 ((∀𝑢𝑦 (𝑔𝑢) = 1o ∧ ∀𝑢 ∈ {𝑧} (𝑔𝑢) = 1o) → ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1o)
5043, 48, 49syl2anc 409 . . . . . . . . . 10 (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ (𝑔𝑧) = 1o) ∧ ∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1o) → ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1o)
5150olcd 729 . . . . . . . . 9 (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ (𝑔𝑧) = 1o) ∧ ∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1o) → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1o))
5251ex 114 . . . . . . . 8 ((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ (𝑔𝑧) = 1o) → (∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1o → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1o)))
53 simpr 109 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) → 𝑔:(𝑦 ∪ {𝑧})⟶2o)
5453, 28ffvelrnd 5629 . . . . . . . . . 10 (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) → (𝑔𝑧) ∈ 2o)
55 df2o3 6406 . . . . . . . . . 10 2o = {∅, 1o}
5654, 55eleqtrdi 2263 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) → (𝑔𝑧) ∈ {∅, 1o})
57 elpri 3604 . . . . . . . . 9 ((𝑔𝑧) ∈ {∅, 1o} → ((𝑔𝑧) = ∅ ∨ (𝑔𝑧) = 1o))
5856, 57syl 14 . . . . . . . 8 (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) → ((𝑔𝑧) = ∅ ∨ (𝑔𝑧) = 1o))
5934, 52, 58mpjaodan 793 . . . . . . 7 (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) → (∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1o → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1o)))
60 vex 2733 . . . . . . . . . . . 12 𝑦 ∈ V
61 isomni 7108 . . . . . . . . . . . 12 (𝑦 ∈ V → (𝑦 ∈ Omni ↔ ∀𝑓(𝑓:𝑦⟶2o → (∃𝑥𝑦 (𝑓𝑥) = ∅ ∨ ∀𝑥𝑦 (𝑓𝑥) = 1o))))
6260, 61ax-mp 5 . . . . . . . . . . 11 (𝑦 ∈ Omni ↔ ∀𝑓(𝑓:𝑦⟶2o → (∃𝑥𝑦 (𝑓𝑥) = ∅ ∨ ∀𝑥𝑦 (𝑓𝑥) = 1o)))
6362biimpi 119 . . . . . . . . . 10 (𝑦 ∈ Omni → ∀𝑓(𝑓:𝑦⟶2o → (∃𝑥𝑦 (𝑓𝑥) = ∅ ∨ ∀𝑥𝑦 (𝑓𝑥) = 1o)))
6463adantl 275 . . . . . . . . 9 ((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) → ∀𝑓(𝑓:𝑦⟶2o → (∃𝑥𝑦 (𝑓𝑥) = ∅ ∨ ∀𝑥𝑦 (𝑓𝑥) = 1o)))
65 vex 2733 . . . . . . . . . . 11 𝑔 ∈ V
6665resex 4930 . . . . . . . . . 10 (𝑔𝑦) ∈ V
67 feq1 5328 . . . . . . . . . . 11 (𝑓 = (𝑔𝑦) → (𝑓:𝑦⟶2o ↔ (𝑔𝑦):𝑦⟶2o))
68 fveq1 5493 . . . . . . . . . . . . . 14 (𝑓 = (𝑔𝑦) → (𝑓𝑥) = ((𝑔𝑦)‘𝑥))
6968eqeq1d 2179 . . . . . . . . . . . . 13 (𝑓 = (𝑔𝑦) → ((𝑓𝑥) = ∅ ↔ ((𝑔𝑦)‘𝑥) = ∅))
7069rexbidv 2471 . . . . . . . . . . . 12 (𝑓 = (𝑔𝑦) → (∃𝑥𝑦 (𝑓𝑥) = ∅ ↔ ∃𝑥𝑦 ((𝑔𝑦)‘𝑥) = ∅))
7168eqeq1d 2179 . . . . . . . . . . . . 13 (𝑓 = (𝑔𝑦) → ((𝑓𝑥) = 1o ↔ ((𝑔𝑦)‘𝑥) = 1o))
7271ralbidv 2470 . . . . . . . . . . . 12 (𝑓 = (𝑔𝑦) → (∀𝑥𝑦 (𝑓𝑥) = 1o ↔ ∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1o))
7370, 72orbi12d 788 . . . . . . . . . . 11 (𝑓 = (𝑔𝑦) → ((∃𝑥𝑦 (𝑓𝑥) = ∅ ∨ ∀𝑥𝑦 (𝑓𝑥) = 1o) ↔ (∃𝑥𝑦 ((𝑔𝑦)‘𝑥) = ∅ ∨ ∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1o)))
7467, 73imbi12d 233 . . . . . . . . . 10 (𝑓 = (𝑔𝑦) → ((𝑓:𝑦⟶2o → (∃𝑥𝑦 (𝑓𝑥) = ∅ ∨ ∀𝑥𝑦 (𝑓𝑥) = 1o)) ↔ ((𝑔𝑦):𝑦⟶2o → (∃𝑥𝑦 ((𝑔𝑦)‘𝑥) = ∅ ∨ ∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1o))))
7566, 74spcv 2824 . . . . . . . . 9 (∀𝑓(𝑓:𝑦⟶2o → (∃𝑥𝑦 (𝑓𝑥) = ∅ ∨ ∀𝑥𝑦 (𝑓𝑥) = 1o)) → ((𝑔𝑦):𝑦⟶2o → (∃𝑥𝑦 ((𝑔𝑦)‘𝑥) = ∅ ∨ ∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1o)))
7664, 75syl 14 . . . . . . . 8 ((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) → ((𝑔𝑦):𝑦⟶2o → (∃𝑥𝑦 ((𝑔𝑦)‘𝑥) = ∅ ∨ ∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1o)))
77 ssun1 3290 . . . . . . . . 9 𝑦 ⊆ (𝑦 ∪ {𝑧})
78 fssres 5371 . . . . . . . . 9 ((𝑔:(𝑦 ∪ {𝑧})⟶2o𝑦 ⊆ (𝑦 ∪ {𝑧})) → (𝑔𝑦):𝑦⟶2o)
7977, 78mpan2 423 . . . . . . . 8 (𝑔:(𝑦 ∪ {𝑧})⟶2o → (𝑔𝑦):𝑦⟶2o)
8076, 79impel 278 . . . . . . 7 (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) → (∃𝑥𝑦 ((𝑔𝑦)‘𝑥) = ∅ ∨ ∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1o))
8124, 59, 80mpjaod 713 . . . . . 6 (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1o))
8281ex 114 . . . . 5 ((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) → (𝑔:(𝑦 ∪ {𝑧})⟶2o → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1o)))
8382alrimiv 1867 . . . 4 ((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) → ∀𝑔(𝑔:(𝑦 ∪ {𝑧})⟶2o → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1o)))
8445snex 4169 . . . . . 6 {𝑧} ∈ V
8560, 84unex 4424 . . . . 5 (𝑦 ∪ {𝑧}) ∈ V
86 isomni 7108 . . . . 5 ((𝑦 ∪ {𝑧}) ∈ V → ((𝑦 ∪ {𝑧}) ∈ Omni ↔ ∀𝑔(𝑔:(𝑦 ∪ {𝑧})⟶2o → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1o))))
8785, 86ax-mp 5 . . . 4 ((𝑦 ∪ {𝑧}) ∈ Omni ↔ ∀𝑔(𝑔:(𝑦 ∪ {𝑧})⟶2o → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1o)))
8883, 87sylibr 133 . . 3 ((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) → (𝑦 ∪ {𝑧}) ∈ Omni)
8988ex 114 . 2 (𝑦 ∈ Fin → (𝑦 ∈ Omni → (𝑦 ∪ {𝑧}) ∈ Omni))
901, 2, 3, 4, 11, 89findcard2 6863 1 (𝐴 ∈ Fin → 𝐴 ∈ Omni)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wo 703  wal 1346   = wceq 1348  wcel 2141  wral 2448  wrex 2449  Vcvv 2730  cun 3119  wss 3121  c0 3414  {csn 3581  {cpr 3582  cres 4611  wf 5192  cfv 5196  1oc1o 6385  2oc2o 6386  Fincfn 6714  Omnicomni 7106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-1o 6392  df-2o 6393  df-er 6509  df-en 6715  df-fin 6717  df-omni 7107
This theorem is referenced by:  trilpolemlt1  13995
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