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Theorem 2ndcsep 22810
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 22797 . 2 (𝐽 ∈ 2ndω ↔ ∃𝑏 ∈ TopBases (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽))
2 vex 3449 . . . . . . . . 9 𝑏 ∈ V
3 difss 4091 . . . . . . . . 9 (𝑏 ∖ {∅}) ⊆ 𝑏
4 ssdomg 8940 . . . . . . . . 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 8947 . . . . . . . 8 (((𝑏 ∖ {∅}) ≼ 𝑏𝑏 ≼ ω) → (𝑏 ∖ {∅}) ≼ ω)
85, 6, 7sylancr 587 . . . . . . 7 ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → (𝑏 ∖ {∅}) ≼ ω)
9 eldifsn 4747 . . . . . . . . 9 (𝑦 ∈ (𝑏 ∖ {∅}) ↔ (𝑦𝑏𝑦 ≠ ∅))
10 n0 4306 . . . . . . . . . 10 (𝑦 ≠ ∅ ↔ ∃𝑧 𝑧𝑦)
11 elunii 4870 . . . . . . . . . . . . . . 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 1920 . . . . . . . . . . . 12 (𝑦𝑏 → (∃𝑧 𝑧𝑦 → ∃𝑧(𝑧 𝑏𝑧𝑦)))
1615imp 407 . . . . . . . . . . 11 ((𝑦𝑏 ∧ ∃𝑧 𝑧𝑦) → ∃𝑧(𝑧 𝑏𝑧𝑦))
17 df-rex 3074 . . . . . . . . . . 11 (∃𝑧 𝑏𝑧𝑦 ↔ ∃𝑧(𝑧 𝑏𝑧𝑦))
1816, 17sylibr 233 . . . . . . . . . 10 ((𝑦𝑏 ∧ ∃𝑧 𝑧𝑦) → ∃𝑧 𝑏𝑧𝑦)
1910, 18sylan2b 594 . . . . . . . . 9 ((𝑦𝑏𝑦 ≠ ∅) → ∃𝑧 𝑏𝑧𝑦)
209, 19sylbi 216 . . . . . . . 8 (𝑦 ∈ (𝑏 ∖ {∅}) → ∃𝑧 𝑏𝑧𝑦)
2120rgen 3066 . . . . . . 7 𝑦 ∈ (𝑏 ∖ {∅})∃𝑧 𝑏𝑧𝑦
22 vuniex 7676 . . . . . . . 8 𝑏 ∈ V
23 eleq1 2825 . . . . . . . 8 (𝑧 = (𝑓𝑦) → (𝑧𝑦 ↔ (𝑓𝑦) ∈ 𝑦))
2422, 23axcc4dom 10377 . . . . . . 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 6675 . . . . . . . . 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 3449 . . . . . . . . . 10 𝑓 ∈ V
2928rnex 7849 . . . . . . . . 9 ran 𝑓 ∈ V
3029elpw 4564 . . . . . . . 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 9582 . . . . . . . . . . 11 ω ∈ On
336adantr 481 . . . . . . . . . . 11 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → 𝑏 ≼ ω)
34 ondomen 9973 . . . . . . . . . . 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 9975 . . . . . . . . . 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 6668 . . . . . . . . . . 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 6762 . . . . . . . . . 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 9993 . . . . . . . . 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 8947 . . . . . . . 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 22319 . . . . . . . . . 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 22317 . . . . . . . . . . . 12 (𝑏 ∈ V → (topGen‘𝑏) = 𝑏)
5049elv 3451 . . . . . . . . . . 11 (topGen‘𝑏) = 𝑏
5150eqcomi 2745 . . . . . . . . . 10 𝑏 = (topGen‘𝑏)
5251clsss3 22410 . . . . . . . . 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 4294 . . . . . . . . . . . . . . . 16 (𝑥𝑦𝑦 ≠ ∅)
5554anim2i 617 . . . . . . . . . . . . . . 15 ((𝑦𝑏𝑥𝑦) → (𝑦𝑏𝑦 ≠ ∅))
5655, 9sylibr 233 . . . . . . . . . . . . . 14 ((𝑦𝑏𝑥𝑦) → 𝑦 ∈ (𝑏 ∖ {∅}))
57 fnfvelrn 7031 . . . . . . . . . . . . . . . . . 18 ((𝑓 Fn (𝑏 ∖ {∅}) ∧ 𝑦 ∈ (𝑏 ∖ {∅})) → (𝑓𝑦) ∈ ran 𝑓)
5838, 57sylan 580 . . . . . . . . . . . . . . . . 17 ((𝑓:(𝑏 ∖ {∅})⟶ 𝑏𝑦 ∈ (𝑏 ∖ {∅})) → (𝑓𝑦) ∈ ran 𝑓)
59 inelcm 4424 . . . . . . . . . . . . . . . . . 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 3160 . . . . . . . . . . 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 725 . . . . . . . . 9 ((((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) ∧ 𝑥 𝑏) → ∀𝑦𝑏 (𝑥𝑦 → (𝑦 ∩ ran 𝑓) ≠ ∅))
69 eqidd 2737 . . . . . . . . . 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 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 22434 . . . . . . . . 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 3965 . . . . . . 7 (((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) ∧ (𝑓:(𝑏 ∖ {∅})⟶ 𝑏 ∧ ∀𝑦 ∈ (𝑏 ∖ {∅})(𝑓𝑦) ∈ 𝑦)) → ((cls‘(topGen‘𝑏))‘ran 𝑓) = 𝑏)
77 breq1 5108 . . . . . . . . 9 (𝑥 = ran 𝑓 → (𝑥 ≼ ω ↔ ran 𝑓 ≼ ω))
78 fveqeq2 6851 . . . . . . . . 9 (𝑥 = ran 𝑓 → (((cls‘(topGen‘𝑏))‘𝑥) = 𝑏 ↔ ((cls‘(topGen‘𝑏))‘ran 𝑓) = 𝑏))
7977, 78anbi12d 631 . . . . . . . 8 (𝑥 = ran 𝑓 → ((𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = 𝑏) ↔ (ran 𝑓 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘ran 𝑓) = 𝑏)))
8079rspcev 3581 . . . . . . 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 1938 . . . . 5 ((𝑏 ∈ TopBases ∧ 𝑏 ≼ ω) → ∃𝑥 ∈ 𝒫 𝑏(𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = 𝑏))
83 unieq 4876 . . . . . . . 8 ((topGen‘𝑏) = 𝐽 (topGen‘𝑏) = 𝐽)
84 2ndcsep.1 . . . . . . . 8 𝑋 = 𝐽
8583, 51, 843eqtr4g 2801 . . . . . . 7 ((topGen‘𝑏) = 𝐽 𝑏 = 𝑋)
8685pweqd 4577 . . . . . 6 ((topGen‘𝑏) = 𝐽 → 𝒫 𝑏 = 𝒫 𝑋)
87 fveq2 6842 . . . . . . . . 9 ((topGen‘𝑏) = 𝐽 → (cls‘(topGen‘𝑏)) = (cls‘𝐽))
8887fveq1d 6844 . . . . . . . 8 ((topGen‘𝑏) = 𝐽 → ((cls‘(topGen‘𝑏))‘𝑥) = ((cls‘𝐽)‘𝑥))
8988, 85eqeq12d 2752 . . . . . . 7 ((topGen‘𝑏) = 𝐽 → (((cls‘(topGen‘𝑏))‘𝑥) = 𝑏 ↔ ((cls‘𝐽)‘𝑥) = 𝑋))
9089anbi2d 629 . . . . . 6 ((topGen‘𝑏) = 𝐽 → ((𝑥 ≼ ω ∧ ((cls‘(topGen‘𝑏))‘𝑥) = 𝑏) ↔ (𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋)))
9186, 90rexeqbidv 3320 . . . . 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 455 . . 3 ((𝑏 ∈ TopBases ∧ (𝑏 ≼ ω ∧ (topGen‘𝑏) = 𝐽)) → ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋))
9493rexlimiva 3144 . 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 396   = wceq 1541  wex 1781  wcel 2106  wne 2943  wral 3064  wrex 3073  Vcvv 3445  cdif 3907  cin 3909  wss 3910  c0 4282  𝒫 cpw 4560  {csn 4586   cuni 4865   class class class wbr 5105  dom cdm 5633  ran crn 5634  Oncon0 6317   Fn wfn 6491  wf 6492  ontowfo 6494  cfv 6496  ωcom 7802  cdom 8881  cardccrd 9871  topGenctg 17319  Topctop 22242  TopBasesctb 22295  clsccl 22369  2ndωc2ndc 22789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cc 10371
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-acn 9878  df-topgen 17325  df-top 22243  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-2ndc 22791
This theorem is referenced by:  met2ndc  23879
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