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Theorem 2ndcdisj2 21785
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 3091 . . 3 (∃*𝑥𝐴 𝑦𝑥 ↔ ∃*𝑥(𝑥𝐴𝑦𝑥))
21albii 1783 . 2 (∀𝑦∃*𝑥𝐴 𝑦𝑥 ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥))
3 undif2 4303 . . . . . 6 ({∅} ∪ (𝐴 ∖ {∅})) = ({∅} ∪ 𝐴)
4 omex 8899 . . . . . . . 8 ω ∈ V
5 peano1 7415 . . . . . . . . 9 ∅ ∈ ω
6 snssi 4612 . . . . . . . . 9 (∅ ∈ ω → {∅} ⊆ ω)
75, 6ax-mp 5 . . . . . . . 8 {∅} ⊆ ω
8 ssdomg 8351 . . . . . . . 8 (ω ∈ V → ({∅} ⊆ ω → {∅} ≼ ω))
94, 7, 8mp2 9 . . . . . . 7 {∅} ≼ ω
10 id 22 . . . . . . . 8 (𝐽 ∈ 2nd𝜔 → 𝐽 ∈ 2nd𝜔)
11 ssdif 4001 . . . . . . . . 9 (𝐴𝐽 → (𝐴 ∖ {∅}) ⊆ (𝐽 ∖ {∅}))
12 dfss3 3842 . . . . . . . . 9 ((𝐴 ∖ {∅}) ⊆ (𝐽 ∖ {∅}) ↔ ∀𝑥 ∈ (𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅}))
1311, 12sylib 210 . . . . . . . 8 (𝐴𝐽 → ∀𝑥 ∈ (𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅}))
14 eldifi 3988 . . . . . . . . . . 11 (𝑥 ∈ (𝐴 ∖ {∅}) → 𝑥𝐴)
1514anim1i 606 . . . . . . . . . 10 ((𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦𝑥) → (𝑥𝐴𝑦𝑥))
1615moimi 2554 . . . . . . . . 9 (∃*𝑥(𝑥𝐴𝑦𝑥) → ∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦𝑥))
1716alimi 1775 . . . . . . . 8 (∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥) → ∀𝑦∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦𝑥))
18 df-rmo 3091 . . . . . . . . . 10 (∃*𝑥 ∈ (𝐴 ∖ {∅})𝑦𝑥 ↔ ∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦𝑥))
1918albii 1783 . . . . . . . . 9 (∀𝑦∃*𝑥 ∈ (𝐴 ∖ {∅})𝑦𝑥 ↔ ∀𝑦∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦𝑥))
20 2ndcdisj 21784 . . . . . . . . 9 ((𝐽 ∈ 2nd𝜔 ∧ ∀𝑥 ∈ (𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅}) ∧ ∀𝑦∃*𝑥 ∈ (𝐴 ∖ {∅})𝑦𝑥) → (𝐴 ∖ {∅}) ≼ ω)
2119, 20syl3an3br 1389 . . . . . . . 8 ((𝐽 ∈ 2nd𝜔 ∧ ∀𝑥 ∈ (𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅}) ∧ ∀𝑦∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦𝑥)) → (𝐴 ∖ {∅}) ≼ ω)
2210, 13, 17, 21syl3an 1141 . . . . . . 7 ((𝐽 ∈ 2nd𝜔 ∧ 𝐴𝐽 ∧ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥)) → (𝐴 ∖ {∅}) ≼ ω)
23 unctb 9424 . . . . . . 7 (({∅} ≼ ω ∧ (𝐴 ∖ {∅}) ≼ ω) → ({∅} ∪ (𝐴 ∖ {∅})) ≼ ω)
249, 22, 23sylancr 579 . . . . . 6 ((𝐽 ∈ 2nd𝜔 ∧ 𝐴𝐽 ∧ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥)) → ({∅} ∪ (𝐴 ∖ {∅})) ≼ ω)
253, 24syl5eqbrr 4962 . . . . 5 ((𝐽 ∈ 2nd𝜔 ∧ 𝐴𝐽 ∧ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥)) → ({∅} ∪ 𝐴) ≼ ω)
26 ctex 8320 . . . . 5 (({∅} ∪ 𝐴) ≼ ω → ({∅} ∪ 𝐴) ∈ V)
2725, 26syl 17 . . . 4 ((𝐽 ∈ 2nd𝜔 ∧ 𝐴𝐽 ∧ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥)) → ({∅} ∪ 𝐴) ∈ V)
28 ssun2 4033 . . . 4 𝐴 ⊆ ({∅} ∪ 𝐴)
29 ssdomg 8351 . . . 4 (({∅} ∪ 𝐴) ∈ V → (𝐴 ⊆ ({∅} ∪ 𝐴) → 𝐴 ≼ ({∅} ∪ 𝐴)))
3027, 28, 29mpisyl 21 . . 3 ((𝐽 ∈ 2nd𝜔 ∧ 𝐴𝐽 ∧ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥)) → 𝐴 ≼ ({∅} ∪ 𝐴))
31 domtr 8358 . . 3 ((𝐴 ≼ ({∅} ∪ 𝐴) ∧ ({∅} ∪ 𝐴) ≼ ω) → 𝐴 ≼ ω)
3230, 25, 31syl2anc 576 . 2 ((𝐽 ∈ 2nd𝜔 ∧ 𝐴𝐽 ∧ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥)) → 𝐴 ≼ ω)
332, 32syl3an3b 1386 1 ((𝐽 ∈ 2nd𝜔 ∧ 𝐴𝐽 ∧ ∀𝑦∃*𝑥𝐴 𝑦𝑥) → 𝐴 ≼ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3a 1069  wal 1506  wcel 2051  ∃*wmo 2546  wral 3083  ∃*wrmo 3086  Vcvv 3410  cdif 3821  cun 3822  wss 3824  c0 4173  {csn 4436   class class class wbr 4926  ωcom 7395  cdom 8303  2nd𝜔c2ndc 21766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-rep 5046  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278  ax-inf2 8897
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-ral 3088  df-rex 3089  df-reu 3090  df-rmo 3091  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-pss 3840  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-tp 4441  df-op 4443  df-uni 4710  df-int 4747  df-iun 4791  df-br 4927  df-opab 4989  df-mpt 5006  df-tr 5028  df-id 5309  df-eprel 5314  df-po 5323  df-so 5324  df-fr 5363  df-se 5364  df-we 5365  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-pred 5984  df-ord 6030  df-on 6031  df-lim 6032  df-suc 6033  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-isom 6195  df-riota 6936  df-ov 6978  df-oprab 6979  df-mpo 6980  df-om 7396  df-1st 7500  df-2nd 7501  df-wrecs 7749  df-recs 7811  df-rdg 7849  df-1o 7904  df-2o 7905  df-oadd 7908  df-er 8088  df-en 8306  df-dom 8307  df-sdom 8308  df-fin 8309  df-oi 8768  df-dju 9123  df-card 9161  df-topgen 16572  df-2ndc 21768
This theorem is referenced by: (None)
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