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Theorem 2ndcdisj2 23466
Description: Any disjoint collection of open sets in a second-countable space is countable. (Contributed by Mario Carneiro, 21-Mar-2015.) (Proof shortened by Mario Carneiro, 9-Apr-2015.) (Revised by NM, 17-Jun-2017.)
Assertion
Ref Expression
2ndcdisj2 ((𝐽 ∈ 2ndω ∧ 𝐴𝐽 ∧ ∀𝑦∃*𝑥𝐴 𝑦𝑥) → 𝐴 ≼ ω)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐽
Allowed substitution hint:   𝐽(𝑦)

Proof of Theorem 2ndcdisj2
StepHypRef Expression
1 df-rmo 3379 . . 3 (∃*𝑥𝐴 𝑦𝑥 ↔ ∃*𝑥(𝑥𝐴𝑦𝑥))
21albii 1818 . 2 (∀𝑦∃*𝑥𝐴 𝑦𝑥 ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥))
3 undif2 4476 . . . . . 6 ({∅} ∪ (𝐴 ∖ {∅})) = ({∅} ∪ 𝐴)
4 omex 9684 . . . . . . . 8 ω ∈ V
5 peano1 7911 . . . . . . . . 9 ∅ ∈ ω
6 snssi 4807 . . . . . . . . 9 (∅ ∈ ω → {∅} ⊆ ω)
75, 6ax-mp 5 . . . . . . . 8 {∅} ⊆ ω
8 ssdomg 9041 . . . . . . . 8 (ω ∈ V → ({∅} ⊆ ω → {∅} ≼ ω))
94, 7, 8mp2 9 . . . . . . 7 {∅} ≼ ω
10 id 22 . . . . . . . 8 (𝐽 ∈ 2ndω → 𝐽 ∈ 2ndω)
11 ssdif 4143 . . . . . . . . 9 (𝐴𝐽 → (𝐴 ∖ {∅}) ⊆ (𝐽 ∖ {∅}))
12 dfss3 3971 . . . . . . . . 9 ((𝐴 ∖ {∅}) ⊆ (𝐽 ∖ {∅}) ↔ ∀𝑥 ∈ (𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅}))
1311, 12sylib 218 . . . . . . . 8 (𝐴𝐽 → ∀𝑥 ∈ (𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅}))
14 eldifi 4130 . . . . . . . . . . 11 (𝑥 ∈ (𝐴 ∖ {∅}) → 𝑥𝐴)
1514anim1i 615 . . . . . . . . . 10 ((𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦𝑥) → (𝑥𝐴𝑦𝑥))
1615moimi 2544 . . . . . . . . 9 (∃*𝑥(𝑥𝐴𝑦𝑥) → ∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦𝑥))
1716alimi 1810 . . . . . . . 8 (∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥) → ∀𝑦∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦𝑥))
18 df-rmo 3379 . . . . . . . . . 10 (∃*𝑥 ∈ (𝐴 ∖ {∅})𝑦𝑥 ↔ ∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦𝑥))
1918albii 1818 . . . . . . . . 9 (∀𝑦∃*𝑥 ∈ (𝐴 ∖ {∅})𝑦𝑥 ↔ ∀𝑦∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦𝑥))
20 2ndcdisj 23465 . . . . . . . . 9 ((𝐽 ∈ 2ndω ∧ ∀𝑥 ∈ (𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅}) ∧ ∀𝑦∃*𝑥 ∈ (𝐴 ∖ {∅})𝑦𝑥) → (𝐴 ∖ {∅}) ≼ ω)
2119, 20syl3an3br 1409 . . . . . . . 8 ((𝐽 ∈ 2ndω ∧ ∀𝑥 ∈ (𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅}) ∧ ∀𝑦∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦𝑥)) → (𝐴 ∖ {∅}) ≼ ω)
2210, 13, 17, 21syl3an 1160 . . . . . . 7 ((𝐽 ∈ 2ndω ∧ 𝐴𝐽 ∧ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥)) → (𝐴 ∖ {∅}) ≼ ω)
23 unctb 10245 . . . . . . 7 (({∅} ≼ ω ∧ (𝐴 ∖ {∅}) ≼ ω) → ({∅} ∪ (𝐴 ∖ {∅})) ≼ ω)
249, 22, 23sylancr 587 . . . . . 6 ((𝐽 ∈ 2ndω ∧ 𝐴𝐽 ∧ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥)) → ({∅} ∪ (𝐴 ∖ {∅})) ≼ ω)
253, 24eqbrtrrid 5178 . . . . 5 ((𝐽 ∈ 2ndω ∧ 𝐴𝐽 ∧ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥)) → ({∅} ∪ 𝐴) ≼ ω)
26 ctex 9005 . . . . 5 (({∅} ∪ 𝐴) ≼ ω → ({∅} ∪ 𝐴) ∈ V)
2725, 26syl 17 . . . 4 ((𝐽 ∈ 2ndω ∧ 𝐴𝐽 ∧ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥)) → ({∅} ∪ 𝐴) ∈ V)
28 ssun2 4178 . . . 4 𝐴 ⊆ ({∅} ∪ 𝐴)
29 ssdomg 9041 . . . 4 (({∅} ∪ 𝐴) ∈ V → (𝐴 ⊆ ({∅} ∪ 𝐴) → 𝐴 ≼ ({∅} ∪ 𝐴)))
3027, 28, 29mpisyl 21 . . 3 ((𝐽 ∈ 2ndω ∧ 𝐴𝐽 ∧ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥)) → 𝐴 ≼ ({∅} ∪ 𝐴))
31 domtr 9048 . . 3 ((𝐴 ≼ ({∅} ∪ 𝐴) ∧ ({∅} ∪ 𝐴) ≼ ω) → 𝐴 ≼ ω)
3230, 25, 31syl2anc 584 . 2 ((𝐽 ∈ 2ndω ∧ 𝐴𝐽 ∧ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥)) → 𝐴 ≼ ω)
332, 32syl3an3b 1406 1 ((𝐽 ∈ 2ndω ∧ 𝐴𝐽 ∧ ∀𝑦∃*𝑥𝐴 𝑦𝑥) → 𝐴 ≼ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086  wal 1537  wcel 2107  ∃*wmo 2537  wral 3060  ∃*wrmo 3378  Vcvv 3479  cdif 3947  cun 3948  wss 3950  c0 4332  {csn 4625   class class class wbr 5142  ωcom 7888  cdom 8984  2ndωc2ndc 23447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-oi 9551  df-dju 9942  df-card 9980  df-topgen 17489  df-2ndc 23449
This theorem is referenced by: (None)
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