Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  finomni GIF version

Theorem finomni 7005
 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 2200 . 2 (𝑤 = ∅ → (𝑤 ∈ Omni ↔ ∅ ∈ Omni))
2 eleq1 2200 . 2 (𝑤 = 𝑦 → (𝑤 ∈ Omni ↔ 𝑦 ∈ Omni))
3 eleq1 2200 . 2 (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 ∈ Omni ↔ (𝑦 ∪ {𝑧}) ∈ Omni))
4 eleq1 2200 . 2 (𝑤 = 𝐴 → (𝑤 ∈ Omni ↔ 𝐴 ∈ Omni))
5 0ex 4050 . . . 4 ∅ ∈ V
6 isomni 7001 . . . 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 3459 . . . . 5 𝑥 ∈ ∅ (𝑓𝑥) = 1o
98olci 721 . . . 4 (∃𝑥 ∈ ∅ (𝑓𝑥) = ∅ ∨ ∀𝑥 ∈ ∅ (𝑓𝑥) = 1o)
109a1i 9 . . 3 (𝑓:∅⟶2o → (∃𝑥 ∈ ∅ (𝑓𝑥) = ∅ ∨ ∀𝑥 ∈ ∅ (𝑓𝑥) = 1o))
117, 10mpgbir 1429 . 2 ∅ ∈ Omni
12 elun1 3238 . . . . . . . . . . . 12 (𝑥𝑦𝑥 ∈ (𝑦 ∪ {𝑧}))
1312ad2antlr 480 . . . . . . . . . . 11 (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ 𝑥𝑦) ∧ ((𝑔𝑦)‘𝑥) = ∅) → 𝑥 ∈ (𝑦 ∪ {𝑧}))
14 fvres 5438 . . . . . . . . . . . . 13 (𝑥𝑦 → ((𝑔𝑦)‘𝑥) = (𝑔𝑥))
1514ad2antlr 480 . . . . . . . . . . . 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 2172 . . . . . . . . . . 11 (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ 𝑥𝑦) ∧ ((𝑔𝑦)‘𝑥) = ∅) → (𝑔𝑥) = ∅)
18 fveq2 5414 . . . . . . . . . . . . 13 (𝑢 = 𝑥 → (𝑔𝑢) = (𝑔𝑥))
1918eqeq1d 2146 . . . . . . . . . . . 12 (𝑢 = 𝑥 → ((𝑔𝑢) = ∅ ↔ (𝑔𝑥) = ∅))
2019rspcev 2784 . . . . . . . . . . 11 ((𝑥 ∈ (𝑦 ∪ {𝑧}) ∧ (𝑔𝑥) = ∅) → ∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅)
2113, 17, 20syl2anc 408 . . . . . . . . . 10 (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ 𝑥𝑦) ∧ ((𝑔𝑦)‘𝑥) = ∅) → ∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅)
2221orcd 722 . . . . . . . . 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 2547 . . . . . . 7 (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) → (∃𝑥𝑦 ((𝑔𝑦)‘𝑥) = ∅ → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1o)))
25 vsnid 3552 . . . . . . . . . . . . 13 𝑧 ∈ {𝑧}
26 elun2 3239 . . . . . . . . . . . . 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 5414 . . . . . . . . . . . . 13 (𝑢 = 𝑧 → (𝑔𝑢) = (𝑔𝑧))
3029eqeq1d 2146 . . . . . . . . . . . 12 (𝑢 = 𝑧 → ((𝑔𝑢) = ∅ ↔ (𝑔𝑧) = ∅))
3130rspcev 2784 . . . . . . . . . . 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 722 . . . . . . . . 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 5414 . . . . . . . . . . . . . . 15 (𝑥 = 𝑢 → ((𝑔𝑦)‘𝑥) = ((𝑔𝑦)‘𝑢))
3736eqeq1d 2146 . . . . . . . . . . . . . 14 (𝑥 = 𝑢 → (((𝑔𝑦)‘𝑥) = 1o ↔ ((𝑔𝑦)‘𝑢) = 1o))
3837cbvralv 2652 . . . . . . . . . . . . 13 (∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1o ↔ ∀𝑢𝑦 ((𝑔𝑦)‘𝑢) = 1o)
39 fvres 5438 . . . . . . . . . . . . . . 15 (𝑢𝑦 → ((𝑔𝑦)‘𝑢) = (𝑔𝑢))
4039eqeq1d 2146 . . . . . . . . . . . . . 14 (𝑢𝑦 → (((𝑔𝑦)‘𝑢) = 1o ↔ (𝑔𝑢) = 1o))
4140ralbiia 2447 . . . . . . . . . . . . 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 519 . . . . . . . . . . . 12 (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ (𝑔𝑧) = 1o) ∧ ∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1o) → (𝑔𝑧) = 1o)
45 vex 2684 . . . . . . . . . . . . 13 𝑧 ∈ V
4629eqeq1d 2146 . . . . . . . . . . . . 13 (𝑢 = 𝑧 → ((𝑔𝑢) = 1o ↔ (𝑔𝑧) = 1o))
4745, 46ralsn 3562 . . . . . . . . . . . 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 3253 . . . . . . . . . . 11 ((∀𝑢𝑦 (𝑔𝑢) = 1o ∧ ∀𝑢 ∈ {𝑧} (𝑔𝑢) = 1o) → ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1o)
5043, 48, 49syl2anc 408 . . . . . . . . . 10 (((((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) ∧ (𝑔𝑧) = 1o) ∧ ∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1o) → ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1o)
5150olcd 723 . . . . . . . . 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 5549 . . . . . . . . . 10 (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) → (𝑔𝑧) ∈ 2o)
55 df2o3 6320 . . . . . . . . . 10 2o = {∅, 1o}
5654, 55eleqtrdi 2230 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) → (𝑔𝑧) ∈ {∅, 1o})
57 elpri 3545 . . . . . . . . 9 ((𝑔𝑧) ∈ {∅, 1o} → ((𝑔𝑧) = ∅ ∨ (𝑔𝑧) = 1o))
5856, 57syl 14 . . . . . . . 8 (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) → ((𝑔𝑧) = ∅ ∨ (𝑔𝑧) = 1o))
5934, 52, 58mpjaodan 787 . . . . . . 7 (((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) ∧ 𝑔:(𝑦 ∪ {𝑧})⟶2o) → (∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1o → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1o)))
60 vex 2684 . . . . . . . . . . . 12 𝑦 ∈ V
61 isomni 7001 . . . . . . . . . . . 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 2684 . . . . . . . . . . 11 𝑔 ∈ V
6665resex 4855 . . . . . . . . . 10 (𝑔𝑦) ∈ V
67 feq1 5250 . . . . . . . . . . 11 (𝑓 = (𝑔𝑦) → (𝑓:𝑦⟶2o ↔ (𝑔𝑦):𝑦⟶2o))
68 fveq1 5413 . . . . . . . . . . . . . 14 (𝑓 = (𝑔𝑦) → (𝑓𝑥) = ((𝑔𝑦)‘𝑥))
6968eqeq1d 2146 . . . . . . . . . . . . 13 (𝑓 = (𝑔𝑦) → ((𝑓𝑥) = ∅ ↔ ((𝑔𝑦)‘𝑥) = ∅))
7069rexbidv 2436 . . . . . . . . . . . 12 (𝑓 = (𝑔𝑦) → (∃𝑥𝑦 (𝑓𝑥) = ∅ ↔ ∃𝑥𝑦 ((𝑔𝑦)‘𝑥) = ∅))
7168eqeq1d 2146 . . . . . . . . . . . . 13 (𝑓 = (𝑔𝑦) → ((𝑓𝑥) = 1o ↔ ((𝑔𝑦)‘𝑥) = 1o))
7271ralbidv 2435 . . . . . . . . . . . 12 (𝑓 = (𝑔𝑦) → (∀𝑥𝑦 (𝑓𝑥) = 1o ↔ ∀𝑥𝑦 ((𝑔𝑦)‘𝑥) = 1o))
7370, 72orbi12d 782 . . . . . . . . . . 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 2774 . . . . . . . . 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 3234 . . . . . . . . 9 𝑦 ⊆ (𝑦 ∪ {𝑧})
78 fssres 5293 . . . . . . . . 9 ((𝑔:(𝑦 ∪ {𝑧})⟶2o𝑦 ⊆ (𝑦 ∪ {𝑧})) → (𝑔𝑦):𝑦⟶2o)
7977, 78mpan2 421 . . . . . . . 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 707 . . . . . 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 1846 . . . 4 ((𝑦 ∈ Fin ∧ 𝑦 ∈ Omni) → ∀𝑔(𝑔:(𝑦 ∪ {𝑧})⟶2o → (∃𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = ∅ ∨ ∀𝑢 ∈ (𝑦 ∪ {𝑧})(𝑔𝑢) = 1o)))
8445snex 4104 . . . . . 6 {𝑧} ∈ V
8560, 84unex 4357 . . . . 5 (𝑦 ∪ {𝑧}) ∈ V
86 isomni 7001 . . . . 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 6776 1 (𝐴 ∈ Fin → 𝐴 ∈ Omni)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697  ∀wal 1329   = wceq 1331   ∈ wcel 1480  ∀wral 2414  ∃wrex 2415  Vcvv 2681   ∪ cun 3064   ⊆ wss 3066  ∅c0 3358  {csn 3522  {cpr 3523   ↾ cres 4536  ⟶wf 5114  ‘cfv 5118  1oc1o 6299  2oc2o 6300  Fincfn 6627  Omnicomni 6997 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497 This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-1o 6306  df-2o 6307  df-er 6422  df-en 6628  df-fin 6630  df-omni 6999 This theorem is referenced by:  trilpolemlt1  13223
 Copyright terms: Public domain W3C validator