Theorem fin23lem11 9592

Description: Lemma for isfin2-2 9594. (Contributed by Stefan O'Rear, 31-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)

Hypotheses

Ref	Expression
fin23lem11.1	⊢ (𝑧 = (𝐴 ∖ 𝑥) → (𝜓 ↔ 𝜒))
fin23lem11.2	⊢ (𝑤 = (𝐴 ∖ 𝑣) → (𝜑 ↔ 𝜃))
fin23lem11.3	⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑣 ⊆ 𝐴) → (𝜒 ↔ 𝜃))

Assertion

Ref	Expression
fin23lem11	⊢ (𝐵 ⊆ 𝒫 𝐴 → (∃𝑥 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵}∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝜑 → ∃𝑧 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ¬ 𝜓))

Distinct variable groups:   𝑣,𝑐,𝑤,𝑥,𝑧,𝐴   𝐵,𝑐,𝑣,𝑤,𝑥,𝑧   𝜒,𝑧   𝜑,𝑣   𝜓,𝑥   𝜃,𝑤

Allowed substitution hints:   𝜑(𝑥,𝑧,𝑤,𝑐)   𝜓(𝑧,𝑤,𝑣,𝑐)   𝜒(𝑥,𝑤,𝑣,𝑐)   𝜃(𝑥,𝑧,𝑣,𝑐)

Proof of Theorem fin23lem11

Step	Hyp	Ref	Expression
1		difeq2 4020	. . . . 5 ⊢ (𝑐 = 𝑥 → (𝐴 ∖ 𝑐) = (𝐴 ∖ 𝑥))
2	1	eleq1d 2869	. . . 4 ⊢ (𝑐 = 𝑥 → ((𝐴 ∖ 𝑐) ∈ 𝐵 ↔ (𝐴 ∖ 𝑥) ∈ 𝐵))
3	2	elrab 3621	. . 3 ⊢ (𝑥 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝐵))
4		simp2r 1193	. . . . 5 ⊢ ((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝜑) → (𝐴 ∖ 𝑥) ∈ 𝐵)
5		fin23lem11.2	. . . . . . . . 9 ⊢ (𝑤 = (𝐴 ∖ 𝑣) → (𝜑 ↔ 𝜃))
6	5	notbid 319	. . . . . . . 8 ⊢ (𝑤 = (𝐴 ∖ 𝑣) → (¬ 𝜑 ↔ ¬ 𝜃))
7		simpl3 1186	. . . . . . . 8 ⊢ (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣 ∈ 𝐵) → ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝜑)
8		difeq2 4020	. . . . . . . . . 10 ⊢ (𝑐 = (𝐴 ∖ 𝑣) → (𝐴 ∖ 𝑐) = (𝐴 ∖ (𝐴 ∖ 𝑣)))
9	8	eleq1d 2869	. . . . . . . . 9 ⊢ (𝑐 = (𝐴 ∖ 𝑣) → ((𝐴 ∖ 𝑐) ∈ 𝐵 ↔ (𝐴 ∖ (𝐴 ∖ 𝑣)) ∈ 𝐵))
10		difss 4035	. . . . . . . . . 10 ⊢ (𝐴 ∖ 𝑣) ⊆ 𝐴
11		ssun1 4075	. . . . . . . . . . . . 13 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝑥)
12		undif1 4344	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ 𝑥) ∪ 𝑥) = (𝐴 ∪ 𝑥)
13	11, 12	sseqtr4i 3931	. . . . . . . . . . . 12 ⊢ 𝐴 ⊆ ((𝐴 ∖ 𝑥) ∪ 𝑥)
14		simpl2r 1220	. . . . . . . . . . . . 13 ⊢ (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣 ∈ 𝐵) → (𝐴 ∖ 𝑥) ∈ 𝐵)
15		simpl2l 1219	. . . . . . . . . . . . 13 ⊢ (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣 ∈ 𝐵) → 𝑥 ∈ 𝒫 𝐴)
16		unexg 7336	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∖ 𝑥) ∈ 𝐵 ∧ 𝑥 ∈ 𝒫 𝐴) → ((𝐴 ∖ 𝑥) ∪ 𝑥) ∈ V)
17	14, 15, 16	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣 ∈ 𝐵) → ((𝐴 ∖ 𝑥) ∪ 𝑥) ∈ V)
18		ssexg 5125	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ((𝐴 ∖ 𝑥) ∪ 𝑥) ∧ ((𝐴 ∖ 𝑥) ∪ 𝑥) ∈ V) → 𝐴 ∈ V)
19	13, 17, 18	sylancr 587	. . . . . . . . . . 11 ⊢ (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣 ∈ 𝐵) → 𝐴 ∈ V)
20		elpw2g 5145	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → ((𝐴 ∖ 𝑣) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ 𝑣) ⊆ 𝐴))
21	19, 20	syl 17	. . . . . . . . . 10 ⊢ (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣 ∈ 𝐵) → ((𝐴 ∖ 𝑣) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ 𝑣) ⊆ 𝐴))
22	10, 21	mpbiri 259	. . . . . . . . 9 ⊢ (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣 ∈ 𝐵) → (𝐴 ∖ 𝑣) ∈ 𝒫 𝐴)
23		simpl1 1184	. . . . . . . . . . . . 13 ⊢ (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣 ∈ 𝐵) → 𝐵 ⊆ 𝒫 𝐴)
24		simpr 485	. . . . . . . . . . . . 13 ⊢ (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣 ∈ 𝐵) → 𝑣 ∈ 𝐵)
25	23, 24	sseldd 3896	. . . . . . . . . . . 12 ⊢ (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣 ∈ 𝐵) → 𝑣 ∈ 𝒫 𝐴)
26	25	elpwid 4471	. . . . . . . . . . 11 ⊢ (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣 ∈ 𝐵) → 𝑣 ⊆ 𝐴)
27		dfss4 4161	. . . . . . . . . . 11 ⊢ (𝑣 ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ 𝑣)) = 𝑣)
28	26, 27	sylib 219	. . . . . . . . . 10 ⊢ (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣 ∈ 𝐵) → (𝐴 ∖ (𝐴 ∖ 𝑣)) = 𝑣)
29	28, 24	eqeltrd 2885	. . . . . . . . 9 ⊢ (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣 ∈ 𝐵) → (𝐴 ∖ (𝐴 ∖ 𝑣)) ∈ 𝐵)
30	9, 22, 29	elrabd 3623	. . . . . . . 8 ⊢ (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣 ∈ 𝐵) → (𝐴 ∖ 𝑣) ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵})
31	6, 7, 30	rspcdva 3567	. . . . . . 7 ⊢ (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣 ∈ 𝐵) → ¬ 𝜃)
32		simplrl 773	. . . . . . . . . . 11 ⊢ (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝐵)) ∧ 𝑣 ∈ 𝐵) → 𝑥 ∈ 𝒫 𝐴)
33	32	elpwid 4471	. . . . . . . . . 10 ⊢ (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝐵)) ∧ 𝑣 ∈ 𝐵) → 𝑥 ⊆ 𝐴)
34		ssel2 3890	. . . . . . . . . . . 12 ⊢ ((𝐵 ⊆ 𝒫 𝐴 ∧ 𝑣 ∈ 𝐵) → 𝑣 ∈ 𝒫 𝐴)
35	34	adantlr 711	. . . . . . . . . . 11 ⊢ (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝐵)) ∧ 𝑣 ∈ 𝐵) → 𝑣 ∈ 𝒫 𝐴)
36	35	elpwid 4471	. . . . . . . . . 10 ⊢ (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝐵)) ∧ 𝑣 ∈ 𝐵) → 𝑣 ⊆ 𝐴)
37		fin23lem11.3	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑣 ⊆ 𝐴) → (𝜒 ↔ 𝜃))
38	33, 36, 37	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝐵)) ∧ 𝑣 ∈ 𝐵) → (𝜒 ↔ 𝜃))
39	38	notbid 319	. . . . . . . 8 ⊢ (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝐵)) ∧ 𝑣 ∈ 𝐵) → (¬ 𝜒 ↔ ¬ 𝜃))
40	39	3adantl3 1161	. . . . . . 7 ⊢ (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣 ∈ 𝐵) → (¬ 𝜒 ↔ ¬ 𝜃))
41	31, 40	mpbird 258	. . . . . 6 ⊢ (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣 ∈ 𝐵) → ¬ 𝜒)
42	41	ralrimiva 3151	. . . . 5 ⊢ ((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝜑) → ∀𝑣 ∈ 𝐵 ¬ 𝜒)
43		fin23lem11.1	. . . . . . . 8 ⊢ (𝑧 = (𝐴 ∖ 𝑥) → (𝜓 ↔ 𝜒))
44	43	notbid 319	. . . . . . 7 ⊢ (𝑧 = (𝐴 ∖ 𝑥) → (¬ 𝜓 ↔ ¬ 𝜒))
45	44	ralbidv 3166	. . . . . 6 ⊢ (𝑧 = (𝐴 ∖ 𝑥) → (∀𝑣 ∈ 𝐵 ¬ 𝜓 ↔ ∀𝑣 ∈ 𝐵 ¬ 𝜒))
46	45	rspcev 3561	. . . . 5 ⊢ (((𝐴 ∖ 𝑥) ∈ 𝐵 ∧ ∀𝑣 ∈ 𝐵 ¬ 𝜒) → ∃𝑧 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ¬ 𝜓)
47	4, 42, 46	syl2anc 584	. . . 4 ⊢ ((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝜑) → ∃𝑧 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ¬ 𝜓)
48	47	3exp 1112	. . 3 ⊢ (𝐵 ⊆ 𝒫 𝐴 → ((𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝐵) → (∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝜑 → ∃𝑧 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ¬ 𝜓)))
49	3, 48	syl5bi 243	. 2 ⊢ (𝐵 ⊆ 𝒫 𝐴 → (𝑥 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} → (∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝜑 → ∃𝑧 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ¬ 𝜓)))
50	49	rexlimdv 3248	1 ⊢ (𝐵 ⊆ 𝒫 𝐴 → (∃𝑥 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵}∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝜑 → ∃𝑧 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080   = wceq 1525   ∈ wcel 2083  ∀wral 3107  ∃wrex 3108  {crab 3111  Vcvv 3440   ∖ cdif 3862   ∪ cun 3863   ⊆ wss 3865  𝒫 cpw 4459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228  ax-un 7326

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-pw 4461  df-sn 4479  df-pr 4481  df-uni 4752

This theorem is referenced by:  fin2i2  9593  isfin2-2  9594