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Theorem 2ndcsep 23454
Description: A second-countable topology is separable, which is to say it contains a countable dense subset. (Contributed by Mario Carneiro, 13-Apr-2015.)
Hypothesis
Ref Expression
2ndcsep.1 𝑋 = 𝐽
Assertion
Ref Expression
2ndcsep (𝐽 ∈ 2ndω → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))
Distinct variable groups:   𝑥,𝐽   𝑥,𝑋

Proof of Theorem 2ndcsep
Dummy variables 𝑓 𝑏 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 is2ndc 23441 . 2 (𝐽 ∈ 2ndω ↔ ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽))
2 vex 3466 . . . . . . . . 9 𝑏 ∈ V
3 difss 4131 . . . . . . . . 9 (𝑏 ∖ {∅}) ⊆ 𝑏
4 ssdomg 9031 . . . . . . . . 9 (𝑏 ∈ V → ((𝑏 ∖ {∅}) ⊆ 𝑏 → (𝑏 ∖ {∅}) ≼ 𝑏))
52, 3, 4mp2 9 . . . . . . . 8 (𝑏 ∖ {∅}) ≼ 𝑏
6 simpr 483 . . . . . . . 8 ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → 𝑏 ≼ ω)
7 domtr 9038 . . . . . . . 8 (((𝑏 ∖ {∅}) ≼ 𝑏𝑏 ≼ ω) → (𝑏 ∖ {∅}) ≼ ω)
85, 6, 7sylancr 585 . . . . . . 7 ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → (𝑏 ∖ {∅}) ≼ ω)
9 eldifsn 4795 . . . . . . . . 9 (𝑦 ∈ (𝑏 ∖ {∅}) ↔ (𝑦𝑏𝑦 ≠ ∅))
10 n0 4349 . . . . . . . . . 10 (𝑦 ≠ ∅ ↔ ∃𝑧 𝑧𝑦)
11 elunii 4918 . . . . . . . . . . . . . . 15 ((𝑧𝑦𝑦𝑏) → 𝑧 𝑏)
12 simpl 481 . . . . . . . . . . . . . . 15 ((𝑧𝑦𝑦𝑏) → 𝑧𝑦)
1311, 12jca 510 . . . . . . . . . . . . . 14 ((𝑧𝑦𝑦𝑏) → (𝑧 𝑏𝑧𝑦))
1413expcom 412 . . . . . . . . . . . . 13 (𝑦𝑏 → (𝑧𝑦 → (𝑧 𝑏𝑧𝑦)))
1514eximdv 1913 . . . . . . . . . . . 12 (𝑦𝑏 → (∃𝑧 𝑧𝑦 → ∃𝑧(𝑧 𝑏𝑧𝑦)))
1615imp 405 . . . . . . . . . . 11 ((𝑦𝑏 ∧ ∃𝑧 𝑧𝑦) → ∃𝑧(𝑧 𝑏𝑧𝑦))
17 df-rex 3061 . . . . . . . . . . 11 (∃𝑧 𝑏𝑧𝑦 ↔ ∃𝑧(𝑧 𝑏𝑧𝑦))
1816, 17sylibr 233 . . . . . . . . . 10 ((𝑦𝑏 ∧ ∃𝑧 𝑧𝑦) → ∃𝑧 𝑏𝑧𝑦)
1910, 18sylan2b 592 . . . . . . . . 9 ((𝑦𝑏𝑦 ≠ ∅) → ∃𝑧 𝑏𝑧𝑦)
209, 19sylbi 216 . . . . . . . 8 (𝑦 ∈ (𝑏 ∖ {∅}) → ∃𝑧 𝑏𝑧𝑦)
2120rgen 3053 . . . . . . 7 𝑦 ∈ (𝑏 ∖ {∅})∃𝑧 𝑏𝑧𝑦
22 vuniex 7750 . . . . . . . 8 𝑏 ∈ V
23 eleq1 2814 . . . . . . . 8 (𝑧 = (𝑓𝑦) → (𝑧𝑦 ↔ (𝑓𝑦) ∈ 𝑦))
2422, 23axcc4dom 10484 . . . . . . 7 (((𝑏 ∖ {∅}) ≼ ω ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})∃𝑧 𝑏𝑧𝑦) → ∃𝑓(𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦))
258, 21, 24sylancl 584 . . . . . 6 ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → ∃𝑓(𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦))
26 frn 6735 . . . . . . . . 9 (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 → ran 𝑓 𝑏)
2726ad2antrl 726 . . . . . . . 8 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → ran 𝑓 𝑏)
28 vex 3466 . . . . . . . . . 10 𝑓 ∈ V
2928rnex 7923 . . . . . . . . 9 ran 𝑓 ∈ V
3029elpw 4611 . . . . . . . 8 (ran 𝑓 ∈ 𝒫 𝑏 ↔ ran 𝑓 𝑏)
3127, 30sylibr 233 . . . . . . 7 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → ran 𝑓 ∈ 𝒫 𝑏)
32 omelon 9689 . . . . . . . . . . 11 ω ∈ On
336adantr 479 . . . . . . . . . . 11 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → 𝑏 ≼ ω)
34 ondomen 10080 . . . . . . . . . . 11 ((ω ∈ On ∧ 𝑏 ≼ ω) → 𝑏 ∈ dom card)
3532, 33, 34sylancr 585 . . . . . . . . . 10 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → 𝑏 ∈ dom card)
36 ssnum 10082 . . . . . . . . . 10 ((𝑏 ∈ dom card ∧ (𝑏 ∖ {∅}) ⊆ 𝑏) → (𝑏 ∖ {∅}) ∈ dom card)
3735, 3, 36sylancl 584 . . . . . . . . 9 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → (𝑏 ∖ {∅}) ∈ dom card)
38 ffn 6728 . . . . . . . . . . 11 (𝑓:(𝑏 ∖ {∅})⟶ 𝑏𝑓 Fn (𝑏 ∖ {∅}))
3938ad2antrl 726 . . . . . . . . . 10 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → 𝑓 Fn (𝑏 ∖ {∅}))
40 dffn4 6821 . . . . . . . . . 10 (𝑓 Fn (𝑏 ∖ {∅}) ↔ 𝑓:(𝑏 ∖ {∅})–onto→ran 𝑓)
4139, 40sylib 217 . . . . . . . . 9 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → 𝑓:(𝑏 ∖ {∅})–onto→ran 𝑓)
42 fodomnum 10100 . . . . . . . . 9 ((𝑏 ∖ {∅}) ∈ dom card → (𝑓:(𝑏 ∖ {∅})–onto→ran 𝑓 → ran 𝑓 ≼ (𝑏 ∖ {∅})))
4337, 41, 42sylc 65 . . . . . . . 8 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → ran 𝑓 ≼ (𝑏 ∖ {∅}))
448adantr 479 . . . . . . . 8 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → (𝑏 ∖ {∅}) ≼ ω)
45 domtr 9038 . . . . . . . 8 ((ran 𝑓 ≼ (𝑏 ∖ {∅}) ∧ (𝑏 ∖ {∅}) ≼ ω) → ran 𝑓 ≼ ω)
4643, 44, 45syl2anc 582 . . . . . . 7 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → ran 𝑓 ≼ ω)
47 tgcl 22963 . . . . . . . . . 10 (𝑏 ∈ TopBases → (topGen‘𝑏) ∈ Top)
4847ad2antrr 724 . . . . . . . . 9 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → (topGen‘𝑏) ∈ Top)
49 unitg 22961 . . . . . . . . . . . 12 (𝑏 ∈ V → (topGen‘𝑏) = 𝑏)
5049elv 3468 . . . . . . . . . . 11 (topGen‘𝑏) = 𝑏
5150eqcomi 2735 . . . . . . . . . 10 𝑏 = (topGen‘𝑏)
5251clsss3 23054 . . . . . . . . 9 (((topGen‘𝑏) ∈ Top ∧ ran 𝑓 𝑏) → ((cls‘(topGen‘𝑏))‘ran 𝑓) ⊆ 𝑏)
5348, 27, 52syl2anc 582 . . . . . . . 8 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → ((cls‘(topGen‘𝑏))‘ran 𝑓) ⊆ 𝑏)
54 ne0i 4337 . . . . . . . . . . . . . . . 16 (𝑥𝑦𝑦 ≠ ∅)
5554anim2i 615 . . . . . . . . . . . . . . 15 ((𝑦𝑏𝑥𝑦) → (𝑦𝑏𝑦 ≠ ∅))
5655, 9sylibr 233 . . . . . . . . . . . . . 14 ((𝑦𝑏𝑥𝑦) → 𝑦 ∈ (𝑏 ∖ {∅}))
57 fnfvelrn 7094 . . . . . . . . . . . . . . . . . 18 ((𝑓 Fn (𝑏 ∖ {∅}) ∧ 𝑦 ∈ (𝑏 ∖ {∅})) → (𝑓𝑦) ∈ ran 𝑓)
5838, 57sylan 578 . . . . . . . . . . . . . . . . 17 ((𝑓:(𝑏 ∖ {∅})⟶ 𝑏𝑦 ∈ (𝑏 ∖ {∅})) → (𝑓𝑦) ∈ ran 𝑓)
59 inelcm 4469 . . . . . . . . . . . . . . . . . 18 (((𝑓𝑦) ∈ 𝑦 ∧ (𝑓𝑦) ∈ ran 𝑓) → (𝑦 ∩ ran 𝑓) ≠ ∅)
6059expcom 412 . . . . . . . . . . . . . . . . 17 ((𝑓𝑦) ∈ ran 𝑓 → ((𝑓𝑦) ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
6158, 60syl 17 . . . . . . . . . . . . . . . 16 ((𝑓:(𝑏 ∖ {∅})⟶ 𝑏𝑦 ∈ (𝑏 ∖ {∅})) → ((𝑓𝑦) ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
6261ex 411 . . . . . . . . . . . . . . 15 (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 → (𝑦 ∈ (𝑏 ∖ {∅}) → ((𝑓𝑦) ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅)))
6362a2d 29 . . . . . . . . . . . . . 14 (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 → ((𝑦 ∈ (𝑏 ∖ {∅}) → (𝑓𝑦) ∈ 𝑦) → (𝑦 ∈ (𝑏 ∖ {∅}) → (𝑦 ∩ ran 𝑓) ≠ ∅)))
6456, 63syl7 74 . . . . . . . . . . . . 13 (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 → ((𝑦 ∈ (𝑏 ∖ {∅}) → (𝑓𝑦) ∈ 𝑦) → ((𝑦𝑏𝑥𝑦) → (𝑦 ∩ ran 𝑓) ≠ ∅)))
6564exp4a 430 . . . . . . . . . . . 12 (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 → ((𝑦 ∈ (𝑏 ∖ {∅}) → (𝑓𝑦) ∈ 𝑦) → (𝑦𝑏 → (𝑥𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))))
6665ralimdv2 3153 . . . . . . . . . . 11 (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 → (∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦 → ∀𝑦𝑏 (𝑥𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅)))
6766imp 405 . . . . . . . . . 10 ((𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦) → ∀𝑦𝑏 (𝑥𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
6867ad2antlr 725 . . . . . . . . 9 ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) ∧ 𝑥 𝑏) → ∀𝑦𝑏 (𝑥𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
69 eqidd 2727 . . . . . . . . . 10 ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) ∧ 𝑥 𝑏) → (topGen‘𝑏) = (topGen‘𝑏))
7051a1i 11 . . . . . . . . . 10 ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) ∧ 𝑥 𝑏) → 𝑏 = (topGen‘𝑏))
71 simplll 773 . . . . . . . . . 10 ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) ∧ 𝑥 𝑏) → 𝑏 ∈ TopBases)
7227adantr 479 . . . . . . . . . 10 ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) ∧ 𝑥 𝑏) → ran 𝑓 𝑏)
73 simpr 483 . . . . . . . . . 10 ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) ∧ 𝑥 𝑏) → 𝑥 𝑏)
7469, 70, 71, 72, 73elcls3 23078 . . . . . . . . 9 ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) ∧ 𝑥 𝑏) → (𝑥 ∈ ((cls‘(topGen‘𝑏))‘ran 𝑓) ↔ ∀𝑦𝑏 (𝑥𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅)))
7568, 74mpbird 256 . . . . . . . 8 ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) ∧ 𝑥 𝑏) → 𝑥 ∈ ((cls‘(topGen‘𝑏))‘ran 𝑓))
7653, 75eqelssd 4001 . . . . . . 7 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → ((cls‘(topGen‘𝑏))‘ran 𝑓) = 𝑏)
77 breq1 5156 . . . . . . . . 9 (𝑥 = ran 𝑓 → (𝑥 ≼ ω ↔ ran 𝑓 ≼ ω))
78 fveqeq2 6910 . . . . . . . . 9 (𝑥 = ran 𝑓 → (((cls‘(topGen‘𝑏))‘𝑥) = 𝑏 ↔ ((cls‘(topGen‘𝑏))‘ran 𝑓) = 𝑏))
7977, 78anbi12d 630 . . . . . . . 8 (𝑥 = ran 𝑓 → ((𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = 𝑏) ↔ (ran 𝑓 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘ran 𝑓) = 𝑏)))
8079rspcev 3608 . . . . . . 7 ((ran 𝑓 ∈ 𝒫 𝑏 ∧ (ran 𝑓 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘ran 𝑓) = 𝑏)) → ∃𝑥 ∈ 𝒫 𝑏(𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = 𝑏))
8131, 46, 76, 80syl12anc 835 . . . . . 6 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → ∃𝑥 ∈ 𝒫 𝑏(𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = 𝑏))
8225, 81exlimddv 1931 . . . . 5 ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → ∃𝑥 ∈ 𝒫 𝑏(𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = 𝑏))
83 unieq 4924 . . . . . . . 8 ((topGen‘𝑏) = 𝐽 (topGen‘𝑏) = 𝐽)
84 2ndcsep.1 . . . . . . . 8 𝑋 = 𝐽
8583, 51, 843eqtr4g 2791 . . . . . . 7 ((topGen‘𝑏) = 𝐽 𝑏 = 𝑋)
8685pweqd 4624 . . . . . 6 ((topGen‘𝑏) = 𝐽 → 𝒫 𝑏 = 𝒫 𝑋)
87 fveq2 6901 . . . . . . . . 9 ((topGen‘𝑏) = 𝐽 → (cls‘(topGen‘𝑏)) = (cls‘𝐽))
8887fveq1d 6903 . . . . . . . 8 ((topGen‘𝑏) = 𝐽 → ((cls‘(topGen‘𝑏))‘𝑥) = ((cls‘𝐽)‘𝑥))
8988, 85eqeq12d 2742 . . . . . . 7 ((topGen‘𝑏) = 𝐽 → (((cls‘(topGen‘𝑏))‘𝑥) = 𝑏 ↔ ((cls‘𝐽)‘𝑥) = 𝑋))
9089anbi2d 628 . . . . . 6 ((topGen‘𝑏) = 𝐽 → ((𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = 𝑏) ↔ (𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋)))
9186, 90rexeqbidv 3331 . . . . 5 ((topGen‘𝑏) = 𝐽 → (∃𝑥 ∈ 𝒫 𝑏(𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = 𝑏) ↔ ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋)))
9282, 91syl5ibcom 244 . . . 4 ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → ((topGen‘𝑏) = 𝐽 → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋)))
9392impr 453 . . 3 ((𝑏 ∈ TopBases ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))
9493rexlimiva 3137 . 2 (∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽) → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))
951, 94sylbi 216 1 (𝐽 ∈ 2ndω → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wex 1774  wcel 2099  wne 2930  wral 3051  wrex 3060  Vcvv 3462  cdif 3944  cin 3946  wss 3947  c0 4325  𝒫 cpw 4607  {csn 4633   cuni 4913   class class class wbr 5153  dom cdm 5682  ran crn 5683  Oncon0 6376   Fn wfn 6549  wf 6550  ontowfo 6552  cfv 6554  ωcom 7876  cdom 8972  cardccrd 9978  topGenctg 17452  Topctop 22886  TopBasesctb 22939  clsccl 23013  2ndωc2ndc 23433
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 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9684  ax-cc 10478
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-iin 5004  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-er 8734  df-map 8857  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-card 9982  df-acn 9985  df-topgen 17458  df-top 22887  df-bases 22940  df-cld 23014  df-ntr 23015  df-cls 23016  df-2ndc 23435
This theorem is referenced by:  met2ndc  24523
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