Theorem 2ndcdisj2 23399

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 ∧ 𝐴 ⊆ 𝐽 ∧ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) → 𝐴 ≼ ω)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐽

Allowed substitution hint:   𝐽(𝑦)

Proof of Theorem 2ndcdisj2

Step	Hyp	Ref	Expression
1		df-rmo 3348	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥))
2	1	albii 1820	. 2 ⊢ (∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥))
3		undif2 4427	. . . . . 6 ⊢ ({∅} ∪ (𝐴 ∖ {∅})) = ({∅} ∪ 𝐴)
4		omex 9550	. . . . . . . 8 ⊢ ω ∈ V
5		peano1 7829	. . . . . . . . 9 ⊢ ∅ ∈ ω
6		snssi 4762	. . . . . . . . 9 ⊢ (∅ ∈ ω → {∅} ⊆ ω)
7	5, 6	ax-mp 5	. . . . . . . 8 ⊢ {∅} ⊆ ω
8		ssdomg 8935	. . . . . . . 8 ⊢ (ω ∈ V → ({∅} ⊆ ω → {∅} ≼ ω))
9	4, 7, 8	mp2 9	. . . . . . 7 ⊢ {∅} ≼ ω
10		id 22	. . . . . . . 8 ⊢ (𝐽 ∈ 2ndω → 𝐽 ∈ 2ndω)
11		ssdif 4094	. . . . . . . . 9 ⊢ (𝐴 ⊆ 𝐽 → (𝐴 ∖ {∅}) ⊆ (𝐽 ∖ {∅}))
12		dfss3 3920	. . . . . . . . 9 ⊢ ((𝐴 ∖ {∅}) ⊆ (𝐽 ∖ {∅}) ↔ ∀𝑥 ∈ (𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅}))
13	11, 12	sylib 218	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐽 → ∀𝑥 ∈ (𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅}))
14		eldifi 4081	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∖ {∅}) → 𝑥 ∈ 𝐴)
15	14	anim1i 615	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦 ∈ 𝑥) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥))
16	15	moimi 2543	. . . . . . . . 9 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → ∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦 ∈ 𝑥))
17	16	alimi 1812	. . . . . . . 8 ⊢ (∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → ∀𝑦∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦 ∈ 𝑥))
18		df-rmo 3348	. . . . . . . . . 10 ⊢ (∃*𝑥 ∈ (𝐴 ∖ {∅})𝑦 ∈ 𝑥 ↔ ∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦 ∈ 𝑥))
19	18	albii 1820	. . . . . . . . 9 ⊢ (∀𝑦∃*𝑥 ∈ (𝐴 ∖ {∅})𝑦 ∈ 𝑥 ↔ ∀𝑦∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦 ∈ 𝑥))
20		2ndcdisj 23398	. . . . . . . . 9 ⊢ ((𝐽 ∈ 2ndω ∧ ∀𝑥 ∈ (𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅}) ∧ ∀𝑦∃*𝑥 ∈ (𝐴 ∖ {∅})𝑦 ∈ 𝑥) → (𝐴 ∖ {∅}) ≼ ω)
21	19, 20	syl3an3br 1410	. . . . . . . 8 ⊢ ((𝐽 ∈ 2ndω ∧ ∀𝑥 ∈ (𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅}) ∧ ∀𝑦∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦 ∈ 𝑥)) → (𝐴 ∖ {∅}) ≼ ω)
22	10, 13, 17, 21	syl3an 1160	. . . . . . 7 ⊢ ((𝐽 ∈ 2ndω ∧ 𝐴 ⊆ 𝐽 ∧ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → (𝐴 ∖ {∅}) ≼ ω)
23		unctb 10112	. . . . . . 7 ⊢ (({∅} ≼ ω ∧ (𝐴 ∖ {∅}) ≼ ω) → ({∅} ∪ (𝐴 ∖ {∅})) ≼ ω)
24	9, 22, 23	sylancr 587	. . . . . 6 ⊢ ((𝐽 ∈ 2ndω ∧ 𝐴 ⊆ 𝐽 ∧ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → ({∅} ∪ (𝐴 ∖ {∅})) ≼ ω)
25	3, 24	eqbrtrrid 5132	. . . . 5 ⊢ ((𝐽 ∈ 2ndω ∧ 𝐴 ⊆ 𝐽 ∧ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → ({∅} ∪ 𝐴) ≼ ω)
26		ctex 8898	. . . . 5 ⊢ (({∅} ∪ 𝐴) ≼ ω → ({∅} ∪ 𝐴) ∈ V)
27	25, 26	syl 17	. . . 4 ⊢ ((𝐽 ∈ 2ndω ∧ 𝐴 ⊆ 𝐽 ∧ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → ({∅} ∪ 𝐴) ∈ V)
28		ssun2 4129	. . . 4 ⊢ 𝐴 ⊆ ({∅} ∪ 𝐴)
29		ssdomg 8935	. . . 4 ⊢ (({∅} ∪ 𝐴) ∈ V → (𝐴 ⊆ ({∅} ∪ 𝐴) → 𝐴 ≼ ({∅} ∪ 𝐴)))
30	27, 28, 29	mpisyl 21	. . 3 ⊢ ((𝐽 ∈ 2ndω ∧ 𝐴 ⊆ 𝐽 ∧ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → 𝐴 ≼ ({∅} ∪ 𝐴))
31		domtr 8942	. . 3 ⊢ ((𝐴 ≼ ({∅} ∪ 𝐴) ∧ ({∅} ∪ 𝐴) ≼ ω) → 𝐴 ≼ ω)
32	30, 25, 31	syl2anc 584	. 2 ⊢ ((𝐽 ∈ 2ndω ∧ 𝐴 ⊆ 𝐽 ∧ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → 𝐴 ≼ ω)
33	2, 32	syl3an3b 1407	1 ⊢ ((𝐽 ∈ 2ndω ∧ 𝐴 ⊆ 𝐽 ∧ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) → 𝐴 ≼ ω)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086  ∀wal 1539   ∈ wcel 2113  ∃*wmo 2535  ∀wral 3049  ∃*wrmo 3347  Vcvv 3438   ∖ cdif 3896   ∪ cun 3897   ⊆ wss 3899  ∅c0 4283  {csn 4578   class class class wbr 5096  ωcom 7806   ≼ cdom 8879  2^ndωc2ndc 23380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-inf2 9548

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7313  df-ov 7359  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8633  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-oi 9413  df-dju 9811  df-card 9849  df-topgen 17361  df-2ndc 23382

This theorem is referenced by: (None)