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Theorem 2ndcdisj2 22831
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 3352 . . 3 (∃*𝑥𝐴 𝑦𝑥 ↔ ∃*𝑥(𝑥𝐴𝑦𝑥))
21albii 1822 . 2 (∀𝑦∃*𝑥𝐴 𝑦𝑥 ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥))
3 undif2 4440 . . . . . 6 ({∅} ∪ (𝐴 ∖ {∅})) = ({∅} ∪ 𝐴)
4 omex 9587 . . . . . . . 8 ω ∈ V
5 peano1 7829 . . . . . . . . 9 ∅ ∈ ω
6 snssi 4772 . . . . . . . . 9 (∅ ∈ ω → {∅} ⊆ ω)
75, 6ax-mp 5 . . . . . . . 8 {∅} ⊆ ω
8 ssdomg 8946 . . . . . . . 8 (ω ∈ V → ({∅} ⊆ ω → {∅} ≼ ω))
94, 7, 8mp2 9 . . . . . . 7 {∅} ≼ ω
10 id 22 . . . . . . . 8 (𝐽 ∈ 2ndω → 𝐽 ∈ 2ndω)
11 ssdif 4103 . . . . . . . . 9 (𝐴𝐽 → (𝐴 ∖ {∅}) ⊆ (𝐽 ∖ {∅}))
12 dfss3 3936 . . . . . . . . 9 ((𝐴 ∖ {∅}) ⊆ (𝐽 ∖ {∅}) ↔ ∀𝑥 ∈ (𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅}))
1311, 12sylib 217 . . . . . . . 8 (𝐴𝐽 → ∀𝑥 ∈ (𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅}))
14 eldifi 4090 . . . . . . . . . . 11 (𝑥 ∈ (𝐴 ∖ {∅}) → 𝑥𝐴)
1514anim1i 616 . . . . . . . . . 10 ((𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦𝑥) → (𝑥𝐴𝑦𝑥))
1615moimi 2540 . . . . . . . . 9 (∃*𝑥(𝑥𝐴𝑦𝑥) → ∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦𝑥))
1716alimi 1814 . . . . . . . 8 (∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥) → ∀𝑦∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦𝑥))
18 df-rmo 3352 . . . . . . . . . 10 (∃*𝑥 ∈ (𝐴 ∖ {∅})𝑦𝑥 ↔ ∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦𝑥))
1918albii 1822 . . . . . . . . 9 (∀𝑦∃*𝑥 ∈ (𝐴 ∖ {∅})𝑦𝑥 ↔ ∀𝑦∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦𝑥))
20 2ndcdisj 22830 . . . . . . . . 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 1161 . . . . . . 7 ((𝐽 ∈ 2ndω ∧ 𝐴𝐽 ∧ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥)) → (𝐴 ∖ {∅}) ≼ ω)
23 unctb 10149 . . . . . . 7 (({∅} ≼ ω ∧ (𝐴 ∖ {∅}) ≼ ω) → ({∅} ∪ (𝐴 ∖ {∅})) ≼ ω)
249, 22, 23sylancr 588 . . . . . 6 ((𝐽 ∈ 2ndω ∧ 𝐴𝐽 ∧ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥)) → ({∅} ∪ (𝐴 ∖ {∅})) ≼ ω)
253, 24eqbrtrrid 5145 . . . . 5 ((𝐽 ∈ 2ndω ∧ 𝐴𝐽 ∧ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥)) → ({∅} ∪ 𝐴) ≼ ω)
26 ctex 8909 . . . . 5 (({∅} ∪ 𝐴) ≼ ω → ({∅} ∪ 𝐴) ∈ V)
2725, 26syl 17 . . . 4 ((𝐽 ∈ 2ndω ∧ 𝐴𝐽 ∧ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥)) → ({∅} ∪ 𝐴) ∈ V)
28 ssun2 4137 . . . 4 𝐴 ⊆ ({∅} ∪ 𝐴)
29 ssdomg 8946 . . . 4 (({∅} ∪ 𝐴) ∈ V → (𝐴 ⊆ ({∅} ∪ 𝐴) → 𝐴 ≼ ({∅} ∪ 𝐴)))
3027, 28, 29mpisyl 21 . . 3 ((𝐽 ∈ 2ndω ∧ 𝐴𝐽 ∧ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝑥)) → 𝐴 ≼ ({∅} ∪ 𝐴))
31 domtr 8953 . . 3 ((𝐴 ≼ ({∅} ∪ 𝐴) ∧ ({∅} ∪ 𝐴) ≼ ω) → 𝐴 ≼ ω)
3230, 25, 31syl2anc 585 . 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 397  w3a 1088  wal 1540  wcel 2107  ∃*wmo 2533  wral 3061  ∃*wrmo 3351  Vcvv 3447  cdif 3911  cun 3912  wss 3914  c0 4286  {csn 4590   class class class wbr 5109  ωcom 7806  cdom 8887  2ndωc2ndc 22812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-inf2 9585
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-se 5593  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7317  df-ov 7364  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-2o 8417  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-oi 9454  df-dju 9845  df-card 9883  df-topgen 17333  df-2ndc 22814
This theorem is referenced by: (None)
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