Theorem 2ndcdisj2 23401

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 3350	. . 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 4429	. . . . . 6 ⊢ ({∅} ∪ (𝐴 ∖ {∅})) = ({∅} ∪ 𝐴)
4		omex 9552	. . . . . . . 8 ⊢ ω ∈ V
5		peano1 7831	. . . . . . . . 9 ⊢ ∅ ∈ ω
6		snssi 4764	. . . . . . . . 9 ⊢ (∅ ∈ ω → {∅} ⊆ ω)
7	5, 6	ax-mp 5	. . . . . . . 8 ⊢ {∅} ⊆ ω
8		ssdomg 8937	. . . . . . . 8 ⊢ (ω ∈ V → ({∅} ⊆ ω → {∅} ≼ ω))
9	4, 7, 8	mp2 9	. . . . . . 7 ⊢ {∅} ≼ ω
10		id 22	. . . . . . . 8 ⊢ (𝐽 ∈ 2ndω → 𝐽 ∈ 2ndω)
11		ssdif 4096	. . . . . . . . 9 ⊢ (𝐴 ⊆ 𝐽 → (𝐴 ∖ {∅}) ⊆ (𝐽 ∖ {∅}))
12		dfss3 3922	. . . . . . . . 9 ⊢ ((𝐴 ∖ {∅}) ⊆ (𝐽 ∖ {∅}) ↔ ∀𝑥 ∈ (𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅}))
13	11, 12	sylib 218	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐽 → ∀𝑥 ∈ (𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅}))
14		eldifi 4083	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∖ {∅}) → 𝑥 ∈ 𝐴)
15	14	anim1i 615	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦 ∈ 𝑥) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥))
16	15	moimi 2545	. . . . . . . . 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 3350	. . . . . . . . . 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 23400	. . . . . . . . 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 10114	. . . . . . 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 5134	. . . . 5 ⊢ ((𝐽 ∈ 2ndω ∧ 𝐴 ⊆ 𝐽 ∧ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → ({∅} ∪ 𝐴) ≼ ω)
26		ctex 8900	. . . . 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 4131	. . . 4 ⊢ 𝐴 ⊆ ({∅} ∪ 𝐴)
29		ssdomg 8937	. . . 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 8944	. . 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 2537  ∀wral 3051  ∃*wrmo 3349  Vcvv 3440   ∖ cdif 3898   ∪ cun 3899   ⊆ wss 3901  ∅c0 4285  {csn 4580   class class class wbr 5098  ωcom 7808   ≼ cdom 8881  2^ndωc2ndc 23382

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 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-oi 9415  df-dju 9813  df-card 9851  df-topgen 17363  df-2ndc 23384

This theorem is referenced by: (None)