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Theorem 2ndcdisj2 22060
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 3138 . . 3 (∃*𝑥𝐴 𝑦𝑥 ↔ ∃*𝑥(𝑥𝐴𝑦𝑥))
21albii 1821 . 2 (∀𝑦∃*𝑥𝐴 𝑦𝑥 ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥))
3 undif2 4397 . . . . . 6 ({∅} ∪ (𝐴 ∖ {∅})) = ({∅} ∪ 𝐴)
4 omex 9094 . . . . . . . 8 ω ∈ V
5 peano1 7586 . . . . . . . . 9 ∅ ∈ ω
6 snssi 4714 . . . . . . . . 9 (∅ ∈ ω → {∅} ⊆ ω)
75, 6ax-mp 5 . . . . . . . 8 {∅} ⊆ ω
8 ssdomg 8542 . . . . . . . 8 (ω ∈ V → ({∅} ⊆ ω → {∅} ≼ ω))
94, 7, 8mp2 9 . . . . . . 7 {∅} ≼ ω
10 id 22 . . . . . . . 8 (𝐽 ∈ 2ndω → 𝐽 ∈ 2ndω)
11 ssdif 4091 . . . . . . . . 9 (𝐴𝐽 → (𝐴 ∖ {∅}) ⊆ (𝐽 ∖ {∅}))
12 dfss3 3930 . . . . . . . . 9 ((𝐴 ∖ {∅}) ⊆ (𝐽 ∖ {∅}) ↔ ∀𝑥 ∈ (𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅}))
1311, 12sylib 221 . . . . . . . 8 (𝐴𝐽 → ∀𝑥 ∈ (𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅}))
14 eldifi 4078 . . . . . . . . . . 11 (𝑥 ∈ (𝐴 ∖ {∅}) → 𝑥𝐴)
1514anim1i 617 . . . . . . . . . 10 ((𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦𝑥) → (𝑥𝐴𝑦𝑥))
1615moimi 2627 . . . . . . . . 9 (∃*𝑥(𝑥𝐴𝑦𝑥) → ∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦𝑥))
1716alimi 1813 . . . . . . . 8 (∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥) → ∀𝑦∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦𝑥))
18 df-rmo 3138 . . . . . . . . . 10 (∃*𝑥 ∈ (𝐴 ∖ {∅})𝑦𝑥 ↔ ∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦𝑥))
1918albii 1821 . . . . . . . . 9 (∀𝑦∃*𝑥 ∈ (𝐴 ∖ {∅})𝑦𝑥 ↔ ∀𝑦∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦𝑥))
20 2ndcdisj 22059 . . . . . . . . 9 ((𝐽 ∈ 2ndω ∧ ∀𝑥 ∈ (𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅}) ∧ ∀𝑦∃*𝑥 ∈ (𝐴 ∖ {∅})𝑦𝑥) → (𝐴 ∖ {∅}) ≼ ω)
2119, 20syl3an3br 1405 . . . . . . . 8 ((𝐽 ∈ 2ndω ∧ ∀𝑥 ∈ (𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅}) ∧ ∀𝑦∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦𝑥)) → (𝐴 ∖ {∅}) ≼ ω)
2210, 13, 17, 21syl3an 1157 . . . . . . 7 ((𝐽 ∈ 2ndω ∧ 𝐴𝐽 ∧ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥)) → (𝐴 ∖ {∅}) ≼ ω)
23 unctb 9616 . . . . . . 7 (({∅} ≼ ω ∧ (𝐴 ∖ {∅}) ≼ ω) → ({∅} ∪ (𝐴 ∖ {∅})) ≼ ω)
249, 22, 23sylancr 590 . . . . . 6 ((𝐽 ∈ 2ndω ∧ 𝐴𝐽 ∧ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥)) → ({∅} ∪ (𝐴 ∖ {∅})) ≼ ω)
253, 24eqbrtrrid 5078 . . . . 5 ((𝐽 ∈ 2ndω ∧ 𝐴𝐽 ∧ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥)) → ({∅} ∪ 𝐴) ≼ ω)
26 ctex 8511 . . . . 5 (({∅} ∪ 𝐴) ≼ ω → ({∅} ∪ 𝐴) ∈ V)
2725, 26syl 17 . . . 4 ((𝐽 ∈ 2ndω ∧ 𝐴𝐽 ∧ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥)) → ({∅} ∪ 𝐴) ∈ V)
28 ssun2 4124 . . . 4 𝐴 ⊆ ({∅} ∪ 𝐴)
29 ssdomg 8542 . . . 4 (({∅} ∪ 𝐴) ∈ V → (𝐴 ⊆ ({∅} ∪ 𝐴) → 𝐴 ≼ ({∅} ∪ 𝐴)))
3027, 28, 29mpisyl 21 . . 3 ((𝐽 ∈ 2ndω ∧ 𝐴𝐽 ∧ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥)) → 𝐴 ≼ ({∅} ∪ 𝐴))
31 domtr 8549 . . 3 ((𝐴 ≼ ({∅} ∪ 𝐴) ∧ ({∅} ∪ 𝐴) ≼ ω) → 𝐴 ≼ ω)
3230, 25, 31syl2anc 587 . 2 ((𝐽 ∈ 2ndω ∧ 𝐴𝐽 ∧ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥)) → 𝐴 ≼ ω)
332, 32syl3an3b 1402 1 ((𝐽 ∈ 2ndω ∧ 𝐴𝐽 ∧ ∀𝑦∃*𝑥𝐴 𝑦𝑥) → 𝐴 ≼ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084  wal 1536  wcel 2114  ∃*wmo 2620  wral 3130  ∃*wrmo 3133  Vcvv 3469  cdif 3905  cun 3906  wss 3908  c0 4265  {csn 4539   class class class wbr 5042  ωcom 7565  cdom 8494  2ndωc2ndc 22041
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-inf2 9092
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-oi 8962  df-dju 9318  df-card 9356  df-topgen 16708  df-2ndc 22043
This theorem is referenced by: (None)
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