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Theorem epttop 22905
Description: The excluded point topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
Assertion
Ref Expression
epttop ((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ (TopOn‘𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑃
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem epttop
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab 4066 . . . . 5 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ↔ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴)))
2 eleq2 2818 . . . . . . . 8 (𝑥 = 𝑦 → (𝑃𝑥𝑃 𝑦))
3 eqeq1 2732 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 = 𝐴 𝑦 = 𝐴))
42, 3imbi12d 344 . . . . . . 7 (𝑥 = 𝑦 → ((𝑃𝑥𝑥 = 𝐴) ↔ (𝑃 𝑦 𝑦 = 𝐴)))
5 simprl 770 . . . . . . . . 9 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → 𝑦 ⊆ 𝒫 𝐴)
6 sspwuni 5097 . . . . . . . . 9 (𝑦 ⊆ 𝒫 𝐴 𝑦𝐴)
75, 6sylib 217 . . . . . . . 8 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → 𝑦𝐴)
8 vuniex 7738 . . . . . . . . 9 𝑦 ∈ V
98elpw 4602 . . . . . . . 8 ( 𝑦 ∈ 𝒫 𝐴 𝑦𝐴)
107, 9sylibr 233 . . . . . . 7 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → 𝑦 ∈ 𝒫 𝐴)
11 eluni2 4907 . . . . . . . . . 10 (𝑃 𝑦 ↔ ∃𝑥𝑦 𝑃𝑥)
12 r19.29 3110 . . . . . . . . . . . . 13 ((∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴) ∧ ∃𝑥𝑦 𝑃𝑥) → ∃𝑥𝑦 ((𝑃𝑥𝑥 = 𝐴) ∧ 𝑃𝑥))
13 simpr 484 . . . . . . . . . . . . . . . 16 ((𝑥𝑦 ∧ (𝑃𝑥𝑥 = 𝐴)) → (𝑃𝑥𝑥 = 𝐴))
1413impr 454 . . . . . . . . . . . . . . 15 ((𝑥𝑦 ∧ ((𝑃𝑥𝑥 = 𝐴) ∧ 𝑃𝑥)) → 𝑥 = 𝐴)
15 elssuni 4935 . . . . . . . . . . . . . . . 16 (𝑥𝑦𝑥 𝑦)
1615adantr 480 . . . . . . . . . . . . . . 15 ((𝑥𝑦 ∧ ((𝑃𝑥𝑥 = 𝐴) ∧ 𝑃𝑥)) → 𝑥 𝑦)
1714, 16eqsstrrd 4017 . . . . . . . . . . . . . 14 ((𝑥𝑦 ∧ ((𝑃𝑥𝑥 = 𝐴) ∧ 𝑃𝑥)) → 𝐴 𝑦)
1817rexlimiva 3143 . . . . . . . . . . . . 13 (∃𝑥𝑦 ((𝑃𝑥𝑥 = 𝐴) ∧ 𝑃𝑥) → 𝐴 𝑦)
1912, 18syl 17 . . . . . . . . . . . 12 ((∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴) ∧ ∃𝑥𝑦 𝑃𝑥) → 𝐴 𝑦)
2019ex 412 . . . . . . . . . . 11 (∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴) → (∃𝑥𝑦 𝑃𝑥𝐴 𝑦))
2120ad2antll 728 . . . . . . . . . 10 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → (∃𝑥𝑦 𝑃𝑥𝐴 𝑦))
2211, 21biimtrid 241 . . . . . . . . 9 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → (𝑃 𝑦𝐴 𝑦))
2322, 7jctild 525 . . . . . . . 8 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → (𝑃 𝑦 → ( 𝑦𝐴𝐴 𝑦)))
24 eqss 3993 . . . . . . . 8 ( 𝑦 = 𝐴 ↔ ( 𝑦𝐴𝐴 𝑦))
2523, 24imbitrrdi 251 . . . . . . 7 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → (𝑃 𝑦 𝑦 = 𝐴))
264, 10, 25elrabd 3683 . . . . . 6 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)})
2726ex 412 . . . . 5 ((𝐴𝑉𝑃𝐴) → ((𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴)) → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}))
281, 27biimtrid 241 . . . 4 ((𝐴𝑉𝑃𝐴) → (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}))
2928alrimiv 1923 . . 3 ((𝐴𝑉𝑃𝐴) → ∀𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}))
30 inss1 4224 . . . . . . . . 9 (𝑦𝑧) ⊆ 𝑦
31 simprll 778 . . . . . . . . . 10 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → 𝑦 ∈ 𝒫 𝐴)
3231elpwid 4607 . . . . . . . . 9 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → 𝑦𝐴)
3330, 32sstrid 3989 . . . . . . . 8 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → (𝑦𝑧) ⊆ 𝐴)
34 vex 3474 . . . . . . . . . 10 𝑦 ∈ V
3534inex1 5311 . . . . . . . . 9 (𝑦𝑧) ∈ V
3635elpw 4602 . . . . . . . 8 ((𝑦𝑧) ∈ 𝒫 𝐴 ↔ (𝑦𝑧) ⊆ 𝐴)
3733, 36sylibr 233 . . . . . . 7 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → (𝑦𝑧) ∈ 𝒫 𝐴)
38 elin 3961 . . . . . . . 8 (𝑃 ∈ (𝑦𝑧) ↔ (𝑃𝑦𝑃𝑧))
39 simprlr 779 . . . . . . . . . 10 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → (𝑃𝑦𝑦 = 𝐴))
40 simprrr 781 . . . . . . . . . 10 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → (𝑃𝑧𝑧 = 𝐴))
4139, 40anim12d 608 . . . . . . . . 9 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → ((𝑃𝑦𝑃𝑧) → (𝑦 = 𝐴𝑧 = 𝐴)))
42 ineq12 4203 . . . . . . . . . 10 ((𝑦 = 𝐴𝑧 = 𝐴) → (𝑦𝑧) = (𝐴𝐴))
43 inidm 4214 . . . . . . . . . 10 (𝐴𝐴) = 𝐴
4442, 43eqtrdi 2784 . . . . . . . . 9 ((𝑦 = 𝐴𝑧 = 𝐴) → (𝑦𝑧) = 𝐴)
4541, 44syl6 35 . . . . . . . 8 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → ((𝑃𝑦𝑃𝑧) → (𝑦𝑧) = 𝐴))
4638, 45biimtrid 241 . . . . . . 7 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → (𝑃 ∈ (𝑦𝑧) → (𝑦𝑧) = 𝐴))
4737, 46jca 511 . . . . . 6 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → ((𝑦𝑧) ∈ 𝒫 𝐴 ∧ (𝑃 ∈ (𝑦𝑧) → (𝑦𝑧) = 𝐴)))
4847ex 412 . . . . 5 ((𝐴𝑉𝑃𝐴) → (((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴))) → ((𝑦𝑧) ∈ 𝒫 𝐴 ∧ (𝑃 ∈ (𝑦𝑧) → (𝑦𝑧) = 𝐴))))
49 eleq2 2818 . . . . . . . 8 (𝑥 = 𝑦 → (𝑃𝑥𝑃𝑦))
50 eqeq1 2732 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
5149, 50imbi12d 344 . . . . . . 7 (𝑥 = 𝑦 → ((𝑃𝑥𝑥 = 𝐴) ↔ (𝑃𝑦𝑦 = 𝐴)))
5251elrab 3681 . . . . . 6 (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)))
53 eleq2 2818 . . . . . . . 8 (𝑥 = 𝑧 → (𝑃𝑥𝑃𝑧))
54 eqeq1 2732 . . . . . . . 8 (𝑥 = 𝑧 → (𝑥 = 𝐴𝑧 = 𝐴))
5553, 54imbi12d 344 . . . . . . 7 (𝑥 = 𝑧 → ((𝑃𝑥𝑥 = 𝐴) ↔ (𝑃𝑧𝑧 = 𝐴)))
5655elrab 3681 . . . . . 6 (𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))
5752, 56anbi12i 627 . . . . 5 ((𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}) ↔ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴))))
58 eleq2 2818 . . . . . . 7 (𝑥 = (𝑦𝑧) → (𝑃𝑥𝑃 ∈ (𝑦𝑧)))
59 eqeq1 2732 . . . . . . 7 (𝑥 = (𝑦𝑧) → (𝑥 = 𝐴 ↔ (𝑦𝑧) = 𝐴))
6058, 59imbi12d 344 . . . . . 6 (𝑥 = (𝑦𝑧) → ((𝑃𝑥𝑥 = 𝐴) ↔ (𝑃 ∈ (𝑦𝑧) → (𝑦𝑧) = 𝐴)))
6160elrab 3681 . . . . 5 ((𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ↔ ((𝑦𝑧) ∈ 𝒫 𝐴 ∧ (𝑃 ∈ (𝑦𝑧) → (𝑦𝑧) = 𝐴)))
6248, 57, 613imtr4g 296 . . . 4 ((𝐴𝑉𝑃𝐴) → ((𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}) → (𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}))
6362ralrimivv 3194 . . 3 ((𝐴𝑉𝑃𝐴) → ∀𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} (𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)})
64 pwexg 5372 . . . . . 6 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
6564adantr 480 . . . . 5 ((𝐴𝑉𝑃𝐴) → 𝒫 𝐴 ∈ V)
66 rabexg 5327 . . . . 5 (𝒫 𝐴 ∈ V → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ V)
6765, 66syl 17 . . . 4 ((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ V)
68 istopg 22790 . . . 4 ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ V → ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ Top ↔ (∀𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}) ∧ ∀𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} (𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)})))
6967, 68syl 17 . . 3 ((𝐴𝑉𝑃𝐴) → ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ Top ↔ (∀𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}) ∧ ∀𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} (𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)})))
7029, 63, 69mpbir2and 712 . 2 ((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ Top)
71 eleq2 2818 . . . . . 6 (𝑥 = 𝐴 → (𝑃𝑥𝑃𝐴))
72 eqeq1 2732 . . . . . 6 (𝑥 = 𝐴 → (𝑥 = 𝐴𝐴 = 𝐴))
7371, 72imbi12d 344 . . . . 5 (𝑥 = 𝐴 → ((𝑃𝑥𝑥 = 𝐴) ↔ (𝑃𝐴𝐴 = 𝐴)))
74 pwidg 4618 . . . . . 6 (𝐴𝑉𝐴 ∈ 𝒫 𝐴)
7574adantr 480 . . . . 5 ((𝐴𝑉𝑃𝐴) → 𝐴 ∈ 𝒫 𝐴)
76 eqidd 2729 . . . . . 6 ((𝐴𝑉𝑃𝐴) → 𝐴 = 𝐴)
7776a1d 25 . . . . 5 ((𝐴𝑉𝑃𝐴) → (𝑃𝐴𝐴 = 𝐴))
7873, 75, 77elrabd 3683 . . . 4 ((𝐴𝑉𝑃𝐴) → 𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)})
79 elssuni 4935 . . . 4 (𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} → 𝐴 {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)})
8078, 79syl 17 . . 3 ((𝐴𝑉𝑃𝐴) → 𝐴 {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)})
81 ssrab2 4073 . . . . 5 {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ⊆ 𝒫 𝐴
82 sspwuni 5097 . . . . 5 ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ⊆ 𝒫 𝐴 {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ⊆ 𝐴)
8381, 82mpbi 229 . . . 4 {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ⊆ 𝐴
8483a1i 11 . . 3 ((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ⊆ 𝐴)
8580, 84eqssd 3995 . 2 ((𝐴𝑉𝑃𝐴) → 𝐴 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)})
86 istopon 22807 . 2 ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ (TopOn‘𝐴) ↔ ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ Top ∧ 𝐴 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}))
8770, 85, 86sylanbrc 582 1 ((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ (TopOn‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wal 1532   = wceq 1534  wcel 2099  wral 3057  wrex 3066  {crab 3428  Vcvv 3470  cin 3944  wss 3945  𝒫 cpw 4598   cuni 4903  cfv 6542  Topctop 22788  TopOnctopon 22805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-top 22789  df-topon 22806
This theorem is referenced by:  dfac14lem  23514  dfac14  23515
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