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 ∧ 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 3091	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥))
2	1	albii 1783	. 2 ⊢ (∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥))
3		undif2 4303	. . . . . 6 ⊢ ({∅} ∪ (𝐴 ∖ {∅})) = ({∅} ∪ 𝐴)
4		omex 8899	. . . . . . . 8 ⊢ ω ∈ V
5		peano1 7415	. . . . . . . . 9 ⊢ ∅ ∈ ω
6		snssi 4612	. . . . . . . . 9 ⊢ (∅ ∈ ω → {∅} ⊆ ω)
7	5, 6	ax-mp 5	. . . . . . . 8 ⊢ {∅} ⊆ ω
8		ssdomg 8351	. . . . . . . 8 ⊢ (ω ∈ V → ({∅} ⊆ ω → {∅} ≼ ω))
9	4, 7, 8	mp2 9	. . . . . . 7 ⊢ {∅} ≼ ω
10		id 22	. . . . . . . 8 ⊢ (𝐽 ∈ 2nd𝜔 → 𝐽 ∈ 2nd𝜔)
11		ssdif 4001	. . . . . . . . 9 ⊢ (𝐴 ⊆ 𝐽 → (𝐴 ∖ {∅}) ⊆ (𝐽 ∖ {∅}))
12		dfss3 3842	. . . . . . . . 9 ⊢ ((𝐴 ∖ {∅}) ⊆ (𝐽 ∖ {∅}) ↔ ∀𝑥 ∈ (𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅}))
13	11, 12	sylib 210	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐽 → ∀𝑥 ∈ (𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅}))
14		eldifi 3988	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∖ {∅}) → 𝑥 ∈ 𝐴)
15	14	anim1i 606	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦 ∈ 𝑥) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥))
16	15	moimi 2554	. . . . . . . . 9 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → ∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦 ∈ 𝑥))
17	16	alimi 1775	. . . . . . . 8 ⊢ (∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → ∀𝑦∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦 ∈ 𝑥))
18		df-rmo 3091	. . . . . . . . . 10 ⊢ (∃*𝑥 ∈ (𝐴 ∖ {∅})𝑦 ∈ 𝑥 ↔ ∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦 ∈ 𝑥))
19	18	albii 1783	. . . . . . . . 9 ⊢ (∀𝑦∃*𝑥 ∈ (𝐴 ∖ {∅})𝑦 ∈ 𝑥 ↔ ∀𝑦∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦 ∈ 𝑥))
20		2ndcdisj 21784	. . . . . . . . 9 ⊢ ((𝐽 ∈ 2nd𝜔 ∧ ∀𝑥 ∈ (𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅}) ∧ ∀𝑦∃*𝑥 ∈ (𝐴 ∖ {∅})𝑦 ∈ 𝑥) → (𝐴 ∖ {∅}) ≼ ω)
21	19, 20	syl3an3br 1389	. . . . . . . 8 ⊢ ((𝐽 ∈ 2nd𝜔 ∧ ∀𝑥 ∈ (𝐴 ∖ {∅})𝑥 ∈ (𝐽 ∖ {∅}) ∧ ∀𝑦∃*𝑥(𝑥 ∈ (𝐴 ∖ {∅}) ∧ 𝑦 ∈ 𝑥)) → (𝐴 ∖ {∅}) ≼ ω)
22	10, 13, 17, 21	syl3an 1141	. . . . . . 7 ⊢ ((𝐽 ∈ 2nd𝜔 ∧ 𝐴 ⊆ 𝐽 ∧ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → (𝐴 ∖ {∅}) ≼ ω)
23		unctb 9424	. . . . . . 7 ⊢ (({∅} ≼ ω ∧ (𝐴 ∖ {∅}) ≼ ω) → ({∅} ∪ (𝐴 ∖ {∅})) ≼ ω)
24	9, 22, 23	sylancr 579	. . . . . 6 ⊢ ((𝐽 ∈ 2nd𝜔 ∧ 𝐴 ⊆ 𝐽 ∧ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → ({∅} ∪ (𝐴 ∖ {∅})) ≼ ω)
25	3, 24	syl5eqbrr 4962	. . . . 5 ⊢ ((𝐽 ∈ 2nd𝜔 ∧ 𝐴 ⊆ 𝐽 ∧ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → ({∅} ∪ 𝐴) ≼ ω)
26		ctex 8320	. . . . 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 4033	. . . 4 ⊢ 𝐴 ⊆ ({∅} ∪ 𝐴)
29		ssdomg 8351	. . . 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 8358	. . 3 ⊢ ((𝐴 ≼ ({∅} ∪ 𝐴) ∧ ({∅} ∪ 𝐴) ≼ ω) → 𝐴 ≼ ω)
32	30, 25, 31	syl2anc 576	. 2 ⊢ ((𝐽 ∈ 2nd𝜔 ∧ 𝐴 ⊆ 𝐽 ∧ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → 𝐴 ≼ ω)
33	2, 32	syl3an3b 1386	1 ⊢ ((𝐽 ∈ 2nd𝜔 ∧ 𝐴 ⊆ 𝐽 ∧ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) → 𝐴 ≼ ω)

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  2^nd𝜔c2ndc 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)