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Theorem knatar 6564
Description: The Knaster-Tarski theorem says that every monotone function over a complete lattice has a (least) fixpoint. Here we specialize this theorem to the case when the lattice is the powerset lattice 𝒫 𝐴. (Contributed by Mario Carneiro, 11-Jun-2015.)
Hypothesis
Ref Expression
knatar.1 𝑋 = {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹𝑧) ⊆ 𝑧}
Assertion
Ref Expression
knatar ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → (𝑋𝐴 ∧ (𝐹𝑋) = 𝑋))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐹,𝑦,𝑧   𝑥,𝑋,𝑦
Allowed substitution hints:   𝑉(𝑥,𝑦,𝑧)   𝑋(𝑧)

Proof of Theorem knatar
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 knatar.1 . . 3 𝑋 = {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹𝑧) ⊆ 𝑧}
2 pwidg 4146 . . . . 5 (𝐴𝑉𝐴 ∈ 𝒫 𝐴)
323ad2ant1 1080 . . . 4 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → 𝐴 ∈ 𝒫 𝐴)
4 simp2 1060 . . . 4 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → (𝐹𝐴) ⊆ 𝐴)
5 fveq2 6150 . . . . . 6 (𝑧 = 𝐴 → (𝐹𝑧) = (𝐹𝐴))
6 id 22 . . . . . 6 (𝑧 = 𝐴𝑧 = 𝐴)
75, 6sseq12d 3615 . . . . 5 (𝑧 = 𝐴 → ((𝐹𝑧) ⊆ 𝑧 ↔ (𝐹𝐴) ⊆ 𝐴))
87intminss 4470 . . . 4 ((𝐴 ∈ 𝒫 𝐴 ∧ (𝐹𝐴) ⊆ 𝐴) → {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹𝑧) ⊆ 𝑧} ⊆ 𝐴)
93, 4, 8syl2anc 692 . . 3 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹𝑧) ⊆ 𝑧} ⊆ 𝐴)
101, 9syl5eqss 3630 . 2 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → 𝑋𝐴)
11 fveq2 6150 . . . . . . . . . . . . . 14 (𝑧 = 𝑤 → (𝐹𝑧) = (𝐹𝑤))
12 id 22 . . . . . . . . . . . . . 14 (𝑧 = 𝑤𝑧 = 𝑤)
1311, 12sseq12d 3615 . . . . . . . . . . . . 13 (𝑧 = 𝑤 → ((𝐹𝑧) ⊆ 𝑧 ↔ (𝐹𝑤) ⊆ 𝑤))
1413intminss 4470 . . . . . . . . . . . 12 ((𝑤 ∈ 𝒫 𝐴 ∧ (𝐹𝑤) ⊆ 𝑤) → {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹𝑧) ⊆ 𝑧} ⊆ 𝑤)
1514adantl 482 . . . . . . . . . . 11 (((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝐹𝑤) ⊆ 𝑤)) → {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹𝑧) ⊆ 𝑧} ⊆ 𝑤)
161, 15syl5eqss 3630 . . . . . . . . . 10 (((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝐹𝑤) ⊆ 𝑤)) → 𝑋𝑤)
17 vex 3189 . . . . . . . . . . 11 𝑤 ∈ V
1817elpw2 4790 . . . . . . . . . 10 (𝑋 ∈ 𝒫 𝑤𝑋𝑤)
1916, 18sylibr 224 . . . . . . . . 9 (((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝐹𝑤) ⊆ 𝑤)) → 𝑋 ∈ 𝒫 𝑤)
20 simprl 793 . . . . . . . . . 10 (((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝐹𝑤) ⊆ 𝑤)) → 𝑤 ∈ 𝒫 𝐴)
21 simpl3 1064 . . . . . . . . . 10 (((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝐹𝑤) ⊆ 𝑤)) → ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥))
22 pweq 4135 . . . . . . . . . . . 12 (𝑥 = 𝑤 → 𝒫 𝑥 = 𝒫 𝑤)
23 fveq2 6150 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (𝐹𝑥) = (𝐹𝑤))
2423sseq2d 3614 . . . . . . . . . . . 12 (𝑥 = 𝑤 → ((𝐹𝑦) ⊆ (𝐹𝑥) ↔ (𝐹𝑦) ⊆ (𝐹𝑤)))
2522, 24raleqbidv 3141 . . . . . . . . . . 11 (𝑥 = 𝑤 → (∀𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥) ↔ ∀𝑦 ∈ 𝒫 𝑤(𝐹𝑦) ⊆ (𝐹𝑤)))
2625rspcv 3291 . . . . . . . . . 10 (𝑤 ∈ 𝒫 𝐴 → (∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥) → ∀𝑦 ∈ 𝒫 𝑤(𝐹𝑦) ⊆ (𝐹𝑤)))
2720, 21, 26sylc 65 . . . . . . . . 9 (((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝐹𝑤) ⊆ 𝑤)) → ∀𝑦 ∈ 𝒫 𝑤(𝐹𝑦) ⊆ (𝐹𝑤))
28 fveq2 6150 . . . . . . . . . . 11 (𝑦 = 𝑋 → (𝐹𝑦) = (𝐹𝑋))
2928sseq1d 3613 . . . . . . . . . 10 (𝑦 = 𝑋 → ((𝐹𝑦) ⊆ (𝐹𝑤) ↔ (𝐹𝑋) ⊆ (𝐹𝑤)))
3029rspcv 3291 . . . . . . . . 9 (𝑋 ∈ 𝒫 𝑤 → (∀𝑦 ∈ 𝒫 𝑤(𝐹𝑦) ⊆ (𝐹𝑤) → (𝐹𝑋) ⊆ (𝐹𝑤)))
3119, 27, 30sylc 65 . . . . . . . 8 (((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝐹𝑤) ⊆ 𝑤)) → (𝐹𝑋) ⊆ (𝐹𝑤))
32 simprr 795 . . . . . . . 8 (((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝐹𝑤) ⊆ 𝑤)) → (𝐹𝑤) ⊆ 𝑤)
3331, 32sstrd 3594 . . . . . . 7 (((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝐹𝑤) ⊆ 𝑤)) → (𝐹𝑋) ⊆ 𝑤)
3433expr 642 . . . . . 6 (((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) ∧ 𝑤 ∈ 𝒫 𝐴) → ((𝐹𝑤) ⊆ 𝑤 → (𝐹𝑋) ⊆ 𝑤))
3534ralrimiva 2960 . . . . 5 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → ∀𝑤 ∈ 𝒫 𝐴((𝐹𝑤) ⊆ 𝑤 → (𝐹𝑋) ⊆ 𝑤))
36 ssintrab 4467 . . . . 5 ((𝐹𝑋) ⊆ {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹𝑤) ⊆ 𝑤} ↔ ∀𝑤 ∈ 𝒫 𝐴((𝐹𝑤) ⊆ 𝑤 → (𝐹𝑋) ⊆ 𝑤))
3735, 36sylibr 224 . . . 4 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → (𝐹𝑋) ⊆ {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹𝑤) ⊆ 𝑤})
3813cbvrabv 3185 . . . . . 6 {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹𝑧) ⊆ 𝑧} = {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹𝑤) ⊆ 𝑤}
3938inteqi 4446 . . . . 5 {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹𝑧) ⊆ 𝑧} = {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹𝑤) ⊆ 𝑤}
401, 39eqtri 2643 . . . 4 𝑋 = {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹𝑤) ⊆ 𝑤}
4137, 40syl6sseqr 3633 . . 3 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → (𝐹𝑋) ⊆ 𝑋)
423, 10sselpwd 4769 . . . . . . . 8 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → 𝑋 ∈ 𝒫 𝐴)
43 simp3 1061 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥))
44 pweq 4135 . . . . . . . . . . 11 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
45 fveq2 6150 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
4645sseq2d 3614 . . . . . . . . . . 11 (𝑥 = 𝐴 → ((𝐹𝑦) ⊆ (𝐹𝑥) ↔ (𝐹𝑦) ⊆ (𝐹𝐴)))
4744, 46raleqbidv 3141 . . . . . . . . . 10 (𝑥 = 𝐴 → (∀𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥) ↔ ∀𝑦 ∈ 𝒫 𝐴(𝐹𝑦) ⊆ (𝐹𝐴)))
4847rspcv 3291 . . . . . . . . 9 (𝐴 ∈ 𝒫 𝐴 → (∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥) → ∀𝑦 ∈ 𝒫 𝐴(𝐹𝑦) ⊆ (𝐹𝐴)))
493, 43, 48sylc 65 . . . . . . . 8 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → ∀𝑦 ∈ 𝒫 𝐴(𝐹𝑦) ⊆ (𝐹𝐴))
5028sseq1d 3613 . . . . . . . . 9 (𝑦 = 𝑋 → ((𝐹𝑦) ⊆ (𝐹𝐴) ↔ (𝐹𝑋) ⊆ (𝐹𝐴)))
5150rspcv 3291 . . . . . . . 8 (𝑋 ∈ 𝒫 𝐴 → (∀𝑦 ∈ 𝒫 𝐴(𝐹𝑦) ⊆ (𝐹𝐴) → (𝐹𝑋) ⊆ (𝐹𝐴)))
5242, 49, 51sylc 65 . . . . . . 7 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → (𝐹𝑋) ⊆ (𝐹𝐴))
5352, 4sstrd 3594 . . . . . 6 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → (𝐹𝑋) ⊆ 𝐴)
54 fvex 6160 . . . . . . 7 (𝐹𝑋) ∈ V
5554elpw 4138 . . . . . 6 ((𝐹𝑋) ∈ 𝒫 𝐴 ↔ (𝐹𝑋) ⊆ 𝐴)
5653, 55sylibr 224 . . . . 5 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → (𝐹𝑋) ∈ 𝒫 𝐴)
5754elpw 4138 . . . . . . 7 ((𝐹𝑋) ∈ 𝒫 𝑋 ↔ (𝐹𝑋) ⊆ 𝑋)
5841, 57sylibr 224 . . . . . 6 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → (𝐹𝑋) ∈ 𝒫 𝑋)
59 pweq 4135 . . . . . . . . 9 (𝑥 = 𝑋 → 𝒫 𝑥 = 𝒫 𝑋)
60 fveq2 6150 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
6160sseq2d 3614 . . . . . . . . 9 (𝑥 = 𝑋 → ((𝐹𝑦) ⊆ (𝐹𝑥) ↔ (𝐹𝑦) ⊆ (𝐹𝑋)))
6259, 61raleqbidv 3141 . . . . . . . 8 (𝑥 = 𝑋 → (∀𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥) ↔ ∀𝑦 ∈ 𝒫 𝑋(𝐹𝑦) ⊆ (𝐹𝑋)))
6362rspcv 3291 . . . . . . 7 (𝑋 ∈ 𝒫 𝐴 → (∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥) → ∀𝑦 ∈ 𝒫 𝑋(𝐹𝑦) ⊆ (𝐹𝑋)))
6442, 43, 63sylc 65 . . . . . 6 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → ∀𝑦 ∈ 𝒫 𝑋(𝐹𝑦) ⊆ (𝐹𝑋))
65 fveq2 6150 . . . . . . . 8 (𝑦 = (𝐹𝑋) → (𝐹𝑦) = (𝐹‘(𝐹𝑋)))
6665sseq1d 3613 . . . . . . 7 (𝑦 = (𝐹𝑋) → ((𝐹𝑦) ⊆ (𝐹𝑋) ↔ (𝐹‘(𝐹𝑋)) ⊆ (𝐹𝑋)))
6766rspcv 3291 . . . . . 6 ((𝐹𝑋) ∈ 𝒫 𝑋 → (∀𝑦 ∈ 𝒫 𝑋(𝐹𝑦) ⊆ (𝐹𝑋) → (𝐹‘(𝐹𝑋)) ⊆ (𝐹𝑋)))
6858, 64, 67sylc 65 . . . . 5 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → (𝐹‘(𝐹𝑋)) ⊆ (𝐹𝑋))
69 fveq2 6150 . . . . . . 7 (𝑤 = (𝐹𝑋) → (𝐹𝑤) = (𝐹‘(𝐹𝑋)))
70 id 22 . . . . . . 7 (𝑤 = (𝐹𝑋) → 𝑤 = (𝐹𝑋))
7169, 70sseq12d 3615 . . . . . 6 (𝑤 = (𝐹𝑋) → ((𝐹𝑤) ⊆ 𝑤 ↔ (𝐹‘(𝐹𝑋)) ⊆ (𝐹𝑋)))
7271intminss 4470 . . . . 5 (((𝐹𝑋) ∈ 𝒫 𝐴 ∧ (𝐹‘(𝐹𝑋)) ⊆ (𝐹𝑋)) → {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹𝑤) ⊆ 𝑤} ⊆ (𝐹𝑋))
7356, 68, 72syl2anc 692 . . . 4 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → {𝑤 ∈ 𝒫 𝐴 ∣ (𝐹𝑤) ⊆ 𝑤} ⊆ (𝐹𝑋))
7440, 73syl5eqss 3630 . . 3 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → 𝑋 ⊆ (𝐹𝑋))
7541, 74eqssd 3601 . 2 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → (𝐹𝑋) = 𝑋)
7610, 75jca 554 1 ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → (𝑋𝐴 ∧ (𝐹𝑋) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  {crab 2911  wss 3556  𝒫 cpw 4132   cint 4442  cfv 5849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-int 4443  df-br 4616  df-iota 5812  df-fv 5857
This theorem is referenced by: (None)
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