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Theorem 2ndcsep 21476
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 21463 . 2 (𝐽 ∈ 2nd𝜔 ↔ ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽))
2 vex 3394 . . . . . . . . 9 𝑏 ∈ V
3 difss 3936 . . . . . . . . 9 (𝑏 ∖ {∅}) ⊆ 𝑏
4 ssdomg 8238 . . . . . . . . 9 (𝑏 ∈ V → ((𝑏 ∖ {∅}) ⊆ 𝑏 → (𝑏 ∖ {∅}) ≼ 𝑏))
52, 3, 4mp2 9 . . . . . . . 8 (𝑏 ∖ {∅}) ≼ 𝑏
6 simpr 473 . . . . . . . 8 ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → 𝑏 ≼ ω)
7 domtr 8245 . . . . . . . 8 (((𝑏 ∖ {∅}) ≼ 𝑏𝑏 ≼ ω) → (𝑏 ∖ {∅}) ≼ ω)
85, 6, 7sylancr 577 . . . . . . 7 ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → (𝑏 ∖ {∅}) ≼ ω)
9 eldifsn 4508 . . . . . . . . 9 (𝑦 ∈ (𝑏 ∖ {∅}) ↔ (𝑦𝑏𝑦 ≠ ∅))
10 n0 4132 . . . . . . . . . 10 (𝑦 ≠ ∅ ↔ ∃𝑧 𝑧𝑦)
11 elunii 4635 . . . . . . . . . . . . . . 15 ((𝑧𝑦𝑦𝑏) → 𝑧 𝑏)
12 simpl 470 . . . . . . . . . . . . . . 15 ((𝑧𝑦𝑦𝑏) → 𝑧𝑦)
1311, 12jca 503 . . . . . . . . . . . . . 14 ((𝑧𝑦𝑦𝑏) → (𝑧 𝑏𝑧𝑦))
1413expcom 400 . . . . . . . . . . . . 13 (𝑦𝑏 → (𝑧𝑦 → (𝑧 𝑏𝑧𝑦)))
1514eximdv 2008 . . . . . . . . . . . 12 (𝑦𝑏 → (∃𝑧 𝑧𝑦 → ∃𝑧(𝑧 𝑏𝑧𝑦)))
1615imp 395 . . . . . . . . . . 11 ((𝑦𝑏 ∧ ∃𝑧 𝑧𝑦) → ∃𝑧(𝑧 𝑏𝑧𝑦))
17 df-rex 3102 . . . . . . . . . . 11 (∃𝑧 𝑏𝑧𝑦 ↔ ∃𝑧(𝑧 𝑏𝑧𝑦))
1816, 17sylibr 225 . . . . . . . . . 10 ((𝑦𝑏 ∧ ∃𝑧 𝑧𝑦) → ∃𝑧 𝑏𝑧𝑦)
1910, 18sylan2b 583 . . . . . . . . 9 ((𝑦𝑏𝑦 ≠ ∅) → ∃𝑧 𝑏𝑧𝑦)
209, 19sylbi 208 . . . . . . . 8 (𝑦 ∈ (𝑏 ∖ {∅}) → ∃𝑧 𝑏𝑧𝑦)
2120rgen 3110 . . . . . . 7 𝑦 ∈ (𝑏 ∖ {∅})∃𝑧 𝑏𝑧𝑦
22 vuniex 7184 . . . . . . . 8 𝑏 ∈ V
23 eleq1 2873 . . . . . . . 8 (𝑧 = (𝑓𝑦) → (𝑧𝑦 ↔ (𝑓𝑦) ∈ 𝑦))
2422, 23axcc4dom 9548 . . . . . . 7 (((𝑏 ∖ {∅}) ≼ ω ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})∃𝑧 𝑏𝑧𝑦) → ∃𝑓(𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦))
258, 21, 24sylancl 576 . . . . . 6 ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → ∃𝑓(𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦))
26 frn 6262 . . . . . . . . 9 (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 → ran 𝑓 𝑏)
2726ad2antrl 710 . . . . . . . 8 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → ran 𝑓 𝑏)
28 vex 3394 . . . . . . . . . 10 𝑓 ∈ V
2928rnex 7330 . . . . . . . . 9 ran 𝑓 ∈ V
3029elpw 4357 . . . . . . . 8 (ran 𝑓 ∈ 𝒫 𝑏 ↔ ran 𝑓 𝑏)
3127, 30sylibr 225 . . . . . . 7 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → ran 𝑓 ∈ 𝒫 𝑏)
32 omelon 8790 . . . . . . . . . . 11 ω ∈ On
336adantr 468 . . . . . . . . . . 11 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → 𝑏 ≼ ω)
34 ondomen 9143 . . . . . . . . . . 11 ((ω ∈ On ∧ 𝑏 ≼ ω) → 𝑏 ∈ dom card)
3532, 33, 34sylancr 577 . . . . . . . . . 10 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → 𝑏 ∈ dom card)
36 ssnum 9145 . . . . . . . . . 10 ((𝑏 ∈ dom card ∧ (𝑏 ∖ {∅}) ⊆ 𝑏) → (𝑏 ∖ {∅}) ∈ dom card)
3735, 3, 36sylancl 576 . . . . . . . . 9 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → (𝑏 ∖ {∅}) ∈ dom card)
38 ffn 6256 . . . . . . . . . . 11 (𝑓:(𝑏 ∖ {∅})⟶ 𝑏𝑓 Fn (𝑏 ∖ {∅}))
3938ad2antrl 710 . . . . . . . . . 10 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → 𝑓 Fn (𝑏 ∖ {∅}))
40 dffn4 6337 . . . . . . . . . 10 (𝑓 Fn (𝑏 ∖ {∅}) ↔ 𝑓:(𝑏 ∖ {∅})–onto→ran 𝑓)
4139, 40sylib 209 . . . . . . . . 9 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → 𝑓:(𝑏 ∖ {∅})–onto→ran 𝑓)
42 fodomnum 9163 . . . . . . . . 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 468 . . . . . . . 8 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → (𝑏 ∖ {∅}) ≼ ω)
45 domtr 8245 . . . . . . . 8 ((ran 𝑓 ≼ (𝑏 ∖ {∅}) ∧ (𝑏 ∖ {∅}) ≼ ω) → ran 𝑓 ≼ ω)
4643, 44, 45syl2anc 575 . . . . . . 7 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → ran 𝑓 ≼ ω)
47 tgcl 20987 . . . . . . . . . 10 (𝑏 ∈ TopBases → (topGen‘𝑏) ∈ Top)
4847ad2antrr 708 . . . . . . . . 9 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → (topGen‘𝑏) ∈ Top)
49 unitg 20985 . . . . . . . . . . . 12 (𝑏 ∈ V → (topGen‘𝑏) = 𝑏)
502, 49ax-mp 5 . . . . . . . . . . 11 (topGen‘𝑏) = 𝑏
5150eqcomi 2815 . . . . . . . . . 10 𝑏 = (topGen‘𝑏)
5251clsss3 21077 . . . . . . . . 9 (((topGen‘𝑏) ∈ Top ∧ ran 𝑓 𝑏) → ((cls‘(topGen‘𝑏))‘ran 𝑓) ⊆ 𝑏)
5348, 27, 52syl2anc 575 . . . . . . . 8 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → ((cls‘(topGen‘𝑏))‘ran 𝑓) ⊆ 𝑏)
54 ne0i 4122 . . . . . . . . . . . . . . . 16 (𝑥𝑦𝑦 ≠ ∅)
5554anim2i 605 . . . . . . . . . . . . . . 15 ((𝑦𝑏𝑥𝑦) → (𝑦𝑏𝑦 ≠ ∅))
5655, 9sylibr 225 . . . . . . . . . . . . . 14 ((𝑦𝑏𝑥𝑦) → 𝑦 ∈ (𝑏 ∖ {∅}))
57 fnfvelrn 6578 . . . . . . . . . . . . . . . . . 18 ((𝑓 Fn (𝑏 ∖ {∅}) ∧ 𝑦 ∈ (𝑏 ∖ {∅})) → (𝑓𝑦) ∈ ran 𝑓)
5838, 57sylan 571 . . . . . . . . . . . . . . . . 17 ((𝑓:(𝑏 ∖ {∅})⟶ 𝑏𝑦 ∈ (𝑏 ∖ {∅})) → (𝑓𝑦) ∈ ran 𝑓)
59 inelcm 4229 . . . . . . . . . . . . . . . . . 18 (((𝑓𝑦) ∈ 𝑦 ∧ (𝑓𝑦) ∈ ran 𝑓) → (𝑦 ∩ ran 𝑓) ≠ ∅)
6059expcom 400 . . . . . . . . . . . . . . . . 17 ((𝑓𝑦) ∈ ran 𝑓 → ((𝑓𝑦) ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
6158, 60syl 17 . . . . . . . . . . . . . . . 16 ((𝑓:(𝑏 ∖ {∅})⟶ 𝑏𝑦 ∈ (𝑏 ∖ {∅})) → ((𝑓𝑦) ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
6261ex 399 . . . . . . . . . . . . . . 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 420 . . . . . . . . . . . 12 (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 → ((𝑦 ∈ (𝑏 ∖ {∅}) → (𝑓𝑦) ∈ 𝑦) → (𝑦𝑏 → (𝑥𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))))
6665ralimdv2 3149 . . . . . . . . . . 11 (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 → (∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦 → ∀𝑦𝑏 (𝑥𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅)))
6766imp 395 . . . . . . . . . 10 ((𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦) → ∀𝑦𝑏 (𝑥𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
6867ad2antlr 709 . . . . . . . . 9 ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) ∧ 𝑥 𝑏) → ∀𝑦𝑏 (𝑥𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
69 eqidd 2807 . . . . . . . . . 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 782 . . . . . . . . . 10 ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) ∧ 𝑥 𝑏) → 𝑏 ∈ TopBases)
7227adantr 468 . . . . . . . . . 10 ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) ∧ 𝑥 𝑏) → ran 𝑓 𝑏)
73 simpr 473 . . . . . . . . . 10 ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) ∧ 𝑥 𝑏) → 𝑥 𝑏)
7469, 70, 71, 72, 73elcls3 21101 . . . . . . . . 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 248 . . . . . . . 8 ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) ∧ 𝑥 𝑏) → 𝑥 ∈ ((cls‘(topGen‘𝑏))‘ran 𝑓))
7653, 75eqelssd 3819 . . . . . . 7 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → ((cls‘(topGen‘𝑏))‘ran 𝑓) = 𝑏)
77 breq1 4847 . . . . . . . . 9 (𝑥 = ran 𝑓 → (𝑥 ≼ ω ↔ ran 𝑓 ≼ ω))
78 fveqeq2 6417 . . . . . . . . 9 (𝑥 = ran 𝑓 → (((cls‘(topGen‘𝑏))‘𝑥) = 𝑏 ↔ ((cls‘(topGen‘𝑏))‘ran 𝑓) = 𝑏))
7977, 78anbi12d 618 . . . . . . . 8 (𝑥 = ran 𝑓 → ((𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = 𝑏) ↔ (ran 𝑓 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘ran 𝑓) = 𝑏)))
8079rspcev 3502 . . . . . . 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 856 . . . . . 6 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → ∃𝑥 ∈ 𝒫 𝑏(𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = 𝑏))
8225, 81exlimddv 2026 . . . . 5 ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → ∃𝑥 ∈ 𝒫 𝑏(𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = 𝑏))
83 unieq 4638 . . . . . . . 8 ((topGen‘𝑏) = 𝐽 (topGen‘𝑏) = 𝐽)
84 2ndcsep.1 . . . . . . . 8 𝑋 = 𝐽
8583, 51, 843eqtr4g 2865 . . . . . . 7 ((topGen‘𝑏) = 𝐽 𝑏 = 𝑋)
8685pweqd 4356 . . . . . 6 ((topGen‘𝑏) = 𝐽 → 𝒫 𝑏 = 𝒫 𝑋)
87 fveq2 6408 . . . . . . . . 9 ((topGen‘𝑏) = 𝐽 → (cls‘(topGen‘𝑏)) = (cls‘𝐽))
8887fveq1d 6410 . . . . . . . 8 ((topGen‘𝑏) = 𝐽 → ((cls‘(topGen‘𝑏))‘𝑥) = ((cls‘𝐽)‘𝑥))
8988, 85eqeq12d 2821 . . . . . . 7 ((topGen‘𝑏) = 𝐽 → (((cls‘(topGen‘𝑏))‘𝑥) = 𝑏 ↔ ((cls‘𝐽)‘𝑥) = 𝑋))
9089anbi2d 616 . . . . . 6 ((topGen‘𝑏) = 𝐽 → ((𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = 𝑏) ↔ (𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋)))
9186, 90rexeqbidv 3342 . . . . 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 236 . . . 4 ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → ((topGen‘𝑏) = 𝐽 → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋)))
9392impr 444 . . 3 ((𝑏 ∈ TopBases ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))
9493rexlimiva 3216 . 2 (∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽) → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))
951, 94sylbi 208 1 (𝐽 ∈ 2nd𝜔 → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wex 1859  wcel 2156  wne 2978  wral 3096  wrex 3097  Vcvv 3391  cdif 3766  cin 3768  wss 3769  c0 4116  𝒫 cpw 4351  {csn 4370   cuni 4630   class class class wbr 4844  dom cdm 5311  ran crn 5312  Oncon0 5936   Fn wfn 6096  wf 6097  ontowfo 6099  cfv 6101  ωcom 7295  cdom 8190  cardccrd 9044  topGenctg 16303  Topctop 20911  TopBasesctb 20963  clsccl 21036  2nd𝜔c2ndc 21455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179  ax-inf2 8785  ax-cc 9542
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-iin 4715  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-se 5271  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-isom 6110  df-riota 6835  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-om 7296  df-1st 7398  df-2nd 7399  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-1o 7796  df-er 7979  df-map 8094  df-en 8193  df-dom 8194  df-sdom 8195  df-fin 8196  df-card 9048  df-acn 9051  df-topgen 16309  df-top 20912  df-bases 20964  df-cld 21037  df-ntr 21038  df-cls 21039  df-2ndc 21457
This theorem is referenced by:  met2ndc  22541
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