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Theorem 2ndcsep 21995
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 21982 . 2 (𝐽 ∈ 2ndω ↔ ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽))
2 vex 3495 . . . . . . . . 9 𝑏 ∈ V
3 difss 4105 . . . . . . . . 9 (𝑏 ∖ {∅}) ⊆ 𝑏
4 ssdomg 8543 . . . . . . . . 9 (𝑏 ∈ V → ((𝑏 ∖ {∅}) ⊆ 𝑏 → (𝑏 ∖ {∅}) ≼ 𝑏))
52, 3, 4mp2 9 . . . . . . . 8 (𝑏 ∖ {∅}) ≼ 𝑏
6 simpr 485 . . . . . . . 8 ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → 𝑏 ≼ ω)
7 domtr 8550 . . . . . . . 8 (((𝑏 ∖ {∅}) ≼ 𝑏𝑏 ≼ ω) → (𝑏 ∖ {∅}) ≼ ω)
85, 6, 7sylancr 587 . . . . . . 7 ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → (𝑏 ∖ {∅}) ≼ ω)
9 eldifsn 4711 . . . . . . . . 9 (𝑦 ∈ (𝑏 ∖ {∅}) ↔ (𝑦𝑏𝑦 ≠ ∅))
10 n0 4307 . . . . . . . . . 10 (𝑦 ≠ ∅ ↔ ∃𝑧 𝑧𝑦)
11 elunii 4835 . . . . . . . . . . . . . . 15 ((𝑧𝑦𝑦𝑏) → 𝑧 𝑏)
12 simpl 483 . . . . . . . . . . . . . . 15 ((𝑧𝑦𝑦𝑏) → 𝑧𝑦)
1311, 12jca 512 . . . . . . . . . . . . . 14 ((𝑧𝑦𝑦𝑏) → (𝑧 𝑏𝑧𝑦))
1413expcom 414 . . . . . . . . . . . . 13 (𝑦𝑏 → (𝑧𝑦 → (𝑧 𝑏𝑧𝑦)))
1514eximdv 1909 . . . . . . . . . . . 12 (𝑦𝑏 → (∃𝑧 𝑧𝑦 → ∃𝑧(𝑧 𝑏𝑧𝑦)))
1615imp 407 . . . . . . . . . . 11 ((𝑦𝑏 ∧ ∃𝑧 𝑧𝑦) → ∃𝑧(𝑧 𝑏𝑧𝑦))
17 df-rex 3141 . . . . . . . . . . 11 (∃𝑧 𝑏𝑧𝑦 ↔ ∃𝑧(𝑧 𝑏𝑧𝑦))
1816, 17sylibr 235 . . . . . . . . . 10 ((𝑦𝑏 ∧ ∃𝑧 𝑧𝑦) → ∃𝑧 𝑏𝑧𝑦)
1910, 18sylan2b 593 . . . . . . . . 9 ((𝑦𝑏𝑦 ≠ ∅) → ∃𝑧 𝑏𝑧𝑦)
209, 19sylbi 218 . . . . . . . 8 (𝑦 ∈ (𝑏 ∖ {∅}) → ∃𝑧 𝑏𝑧𝑦)
2120rgen 3145 . . . . . . 7 𝑦 ∈ (𝑏 ∖ {∅})∃𝑧 𝑏𝑧𝑦
22 vuniex 7455 . . . . . . . 8 𝑏 ∈ V
23 eleq1 2897 . . . . . . . 8 (𝑧 = (𝑓𝑦) → (𝑧𝑦 ↔ (𝑓𝑦) ∈ 𝑦))
2422, 23axcc4dom 9851 . . . . . . 7 (((𝑏 ∖ {∅}) ≼ ω ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})∃𝑧 𝑏𝑧𝑦) → ∃𝑓(𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦))
258, 21, 24sylancl 586 . . . . . 6 ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → ∃𝑓(𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦))
26 frn 6513 . . . . . . . . 9 (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 → ran 𝑓 𝑏)
2726ad2antrl 724 . . . . . . . 8 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → ran 𝑓 𝑏)
28 vex 3495 . . . . . . . . . 10 𝑓 ∈ V
2928rnex 7606 . . . . . . . . 9 ran 𝑓 ∈ V
3029elpw 4542 . . . . . . . 8 (ran 𝑓 ∈ 𝒫 𝑏 ↔ ran 𝑓 𝑏)
3127, 30sylibr 235 . . . . . . 7 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → ran 𝑓 ∈ 𝒫 𝑏)
32 omelon 9097 . . . . . . . . . . 11 ω ∈ On
336adantr 481 . . . . . . . . . . 11 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → 𝑏 ≼ ω)
34 ondomen 9451 . . . . . . . . . . 11 ((ω ∈ On ∧ 𝑏 ≼ ω) → 𝑏 ∈ dom card)
3532, 33, 34sylancr 587 . . . . . . . . . 10 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → 𝑏 ∈ dom card)
36 ssnum 9453 . . . . . . . . . 10 ((𝑏 ∈ dom card ∧ (𝑏 ∖ {∅}) ⊆ 𝑏) → (𝑏 ∖ {∅}) ∈ dom card)
3735, 3, 36sylancl 586 . . . . . . . . 9 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → (𝑏 ∖ {∅}) ∈ dom card)
38 ffn 6507 . . . . . . . . . . 11 (𝑓:(𝑏 ∖ {∅})⟶ 𝑏𝑓 Fn (𝑏 ∖ {∅}))
3938ad2antrl 724 . . . . . . . . . 10 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → 𝑓 Fn (𝑏 ∖ {∅}))
40 dffn4 6589 . . . . . . . . . 10 (𝑓 Fn (𝑏 ∖ {∅}) ↔ 𝑓:(𝑏 ∖ {∅})–onto→ran 𝑓)
4139, 40sylib 219 . . . . . . . . 9 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → 𝑓:(𝑏 ∖ {∅})–onto→ran 𝑓)
42 fodomnum 9471 . . . . . . . . 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 481 . . . . . . . 8 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → (𝑏 ∖ {∅}) ≼ ω)
45 domtr 8550 . . . . . . . 8 ((ran 𝑓 ≼ (𝑏 ∖ {∅}) ∧ (𝑏 ∖ {∅}) ≼ ω) → ran 𝑓 ≼ ω)
4643, 44, 45syl2anc 584 . . . . . . 7 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → ran 𝑓 ≼ ω)
47 tgcl 21505 . . . . . . . . . 10 (𝑏 ∈ TopBases → (topGen‘𝑏) ∈ Top)
4847ad2antrr 722 . . . . . . . . 9 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → (topGen‘𝑏) ∈ Top)
49 unitg 21503 . . . . . . . . . . . 12 (𝑏 ∈ V → (topGen‘𝑏) = 𝑏)
5049elv 3497 . . . . . . . . . . 11 (topGen‘𝑏) = 𝑏
5150eqcomi 2827 . . . . . . . . . 10 𝑏 = (topGen‘𝑏)
5251clsss3 21595 . . . . . . . . 9 (((topGen‘𝑏) ∈ Top ∧ ran 𝑓 𝑏) → ((cls‘(topGen‘𝑏))‘ran 𝑓) ⊆ 𝑏)
5348, 27, 52syl2anc 584 . . . . . . . 8 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → ((cls‘(topGen‘𝑏))‘ran 𝑓) ⊆ 𝑏)
54 ne0i 4297 . . . . . . . . . . . . . . . 16 (𝑥𝑦𝑦 ≠ ∅)
5554anim2i 616 . . . . . . . . . . . . . . 15 ((𝑦𝑏𝑥𝑦) → (𝑦𝑏𝑦 ≠ ∅))
5655, 9sylibr 235 . . . . . . . . . . . . . 14 ((𝑦𝑏𝑥𝑦) → 𝑦 ∈ (𝑏 ∖ {∅}))
57 fnfvelrn 6840 . . . . . . . . . . . . . . . . . 18 ((𝑓 Fn (𝑏 ∖ {∅}) ∧ 𝑦 ∈ (𝑏 ∖ {∅})) → (𝑓𝑦) ∈ ran 𝑓)
5838, 57sylan 580 . . . . . . . . . . . . . . . . 17 ((𝑓:(𝑏 ∖ {∅})⟶ 𝑏𝑦 ∈ (𝑏 ∖ {∅})) → (𝑓𝑦) ∈ ran 𝑓)
59 inelcm 4410 . . . . . . . . . . . . . . . . . 18 (((𝑓𝑦) ∈ 𝑦 ∧ (𝑓𝑦) ∈ ran 𝑓) → (𝑦 ∩ ran 𝑓) ≠ ∅)
6059expcom 414 . . . . . . . . . . . . . . . . 17 ((𝑓𝑦) ∈ ran 𝑓 → ((𝑓𝑦) ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
6158, 60syl 17 . . . . . . . . . . . . . . . 16 ((𝑓:(𝑏 ∖ {∅})⟶ 𝑏𝑦 ∈ (𝑏 ∖ {∅})) → ((𝑓𝑦) ∈ 𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
6261ex 413 . . . . . . . . . . . . . . 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 432 . . . . . . . . . . . 12 (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 → ((𝑦 ∈ (𝑏 ∖ {∅}) → (𝑓𝑦) ∈ 𝑦) → (𝑦𝑏 → (𝑥𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))))
6665ralimdv2 3173 . . . . . . . . . . 11 (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 → (∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦 → ∀𝑦𝑏 (𝑥𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅)))
6766imp 407 . . . . . . . . . 10 ((𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦) → ∀𝑦𝑏 (𝑥𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
6867ad2antlr 723 . . . . . . . . 9 ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) ∧ 𝑥 𝑏) → ∀𝑦𝑏 (𝑥𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
69 eqidd 2819 . . . . . . . . . 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 771 . . . . . . . . . 10 ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) ∧ 𝑥 𝑏) → 𝑏 ∈ TopBases)
7227adantr 481 . . . . . . . . . 10 ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) ∧ 𝑥 𝑏) → ran 𝑓 𝑏)
73 simpr 485 . . . . . . . . . 10 ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) ∧ 𝑥 𝑏) → 𝑥 𝑏)
7469, 70, 71, 72, 73elcls3 21619 . . . . . . . . 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 258 . . . . . . . 8 ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) ∧ 𝑥 𝑏) → 𝑥 ∈ ((cls‘(topGen‘𝑏))‘ran 𝑓))
7653, 75eqelssd 3985 . . . . . . 7 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → ((cls‘(topGen‘𝑏))‘ran 𝑓) = 𝑏)
77 breq1 5060 . . . . . . . . 9 (𝑥 = ran 𝑓 → (𝑥 ≼ ω ↔ ran 𝑓 ≼ ω))
78 fveqeq2 6672 . . . . . . . . 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 3620 . . . . . . 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 832 . . . . . 6 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → ∃𝑥 ∈ 𝒫 𝑏(𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = 𝑏))
8225, 81exlimddv 1927 . . . . 5 ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → ∃𝑥 ∈ 𝒫 𝑏(𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = 𝑏))
83 unieq 4838 . . . . . . . 8 ((topGen‘𝑏) = 𝐽 (topGen‘𝑏) = 𝐽)
84 2ndcsep.1 . . . . . . . 8 𝑋 = 𝐽
8583, 51, 843eqtr4g 2878 . . . . . . 7 ((topGen‘𝑏) = 𝐽 𝑏 = 𝑋)
8685pweqd 4540 . . . . . 6 ((topGen‘𝑏) = 𝐽 → 𝒫 𝑏 = 𝒫 𝑋)
87 fveq2 6663 . . . . . . . . 9 ((topGen‘𝑏) = 𝐽 → (cls‘(topGen‘𝑏)) = (cls‘𝐽))
8887fveq1d 6665 . . . . . . . 8 ((topGen‘𝑏) = 𝐽 → ((cls‘(topGen‘𝑏))‘𝑥) = ((cls‘𝐽)‘𝑥))
8988, 85eqeq12d 2834 . . . . . . 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 3400 . . . . 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 246 . . . 4 ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → ((topGen‘𝑏) = 𝐽 → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋)))
9392impr 455 . . 3 ((𝑏 ∈ TopBases ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))
9493rexlimiva 3278 . 2 (∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽) → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))
951, 94sylbi 218 1 (𝐽 ∈ 2ndω → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wex 1771  wcel 2105  wne 3013  wral 3135  wrex 3136  Vcvv 3492  cdif 3930  cin 3932  wss 3933  c0 4288  𝒫 cpw 4535  {csn 4557   cuni 4830   class class class wbr 5057  dom cdm 5548  ran crn 5549  Oncon0 6184   Fn wfn 6343  wf 6344  ontowfo 6346  cfv 6348  ωcom 7569  cdom 8495  cardccrd 9352  topGenctg 16699  Topctop 21429  TopBasesctb 21481  clsccl 21554  2ndωc2ndc 21974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cc 9845
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-card 9356  df-acn 9359  df-topgen 16705  df-top 21430  df-bases 21482  df-cld 21555  df-ntr 21556  df-cls 21557  df-2ndc 21976
This theorem is referenced by:  met2ndc  23060
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