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Theorem fin23lem11 9177
 Description: Lemma for isfin2-2 9179. (Contributed by Stefan O'Rear, 31-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)
Hypotheses
Ref Expression
fin23lem11.1 (𝑧 = (𝐴𝑥) → (𝜓𝜒))
fin23lem11.2 (𝑤 = (𝐴𝑣) → (𝜑𝜃))
fin23lem11.3 ((𝑥𝐴𝑣𝐴) → (𝜒𝜃))
Assertion
Ref Expression
fin23lem11 (𝐵 ⊆ 𝒫 𝐴 → (∃𝑥 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵}∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝜑 → ∃𝑧𝐵𝑣𝐵 ¬ 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
StepHypRef Expression
1 difeq2 3755 . . . . 5 (𝑐 = 𝑥 → (𝐴𝑐) = (𝐴𝑥))
21eleq1d 2715 . . . 4 (𝑐 = 𝑥 → ((𝐴𝑐) ∈ 𝐵 ↔ (𝐴𝑥) ∈ 𝐵))
32elrab 3396 . . 3 (𝑥 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝐵))
4 simp2r 1108 . . . . 5 ((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝜑) → (𝐴𝑥) ∈ 𝐵)
5 difss 3770 . . . . . . . . . 10 (𝐴𝑣) ⊆ 𝐴
6 ssun1 3809 . . . . . . . . . . . . 13 𝐴 ⊆ (𝐴𝑥)
7 undif1 4076 . . . . . . . . . . . . 13 ((𝐴𝑥) ∪ 𝑥) = (𝐴𝑥)
86, 7sseqtr4i 3671 . . . . . . . . . . . 12 𝐴 ⊆ ((𝐴𝑥) ∪ 𝑥)
9 simpl2r 1135 . . . . . . . . . . . . 13 (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣𝐵) → (𝐴𝑥) ∈ 𝐵)
10 simpl2l 1134 . . . . . . . . . . . . 13 (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣𝐵) → 𝑥 ∈ 𝒫 𝐴)
11 unexg 7001 . . . . . . . . . . . . 13 (((𝐴𝑥) ∈ 𝐵𝑥 ∈ 𝒫 𝐴) → ((𝐴𝑥) ∪ 𝑥) ∈ V)
129, 10, 11syl2anc 694 . . . . . . . . . . . 12 (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣𝐵) → ((𝐴𝑥) ∪ 𝑥) ∈ V)
13 ssexg 4837 . . . . . . . . . . . 12 ((𝐴 ⊆ ((𝐴𝑥) ∪ 𝑥) ∧ ((𝐴𝑥) ∪ 𝑥) ∈ V) → 𝐴 ∈ V)
148, 12, 13sylancr 696 . . . . . . . . . . 11 (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣𝐵) → 𝐴 ∈ V)
15 elpw2g 4857 . . . . . . . . . . 11 (𝐴 ∈ V → ((𝐴𝑣) ∈ 𝒫 𝐴 ↔ (𝐴𝑣) ⊆ 𝐴))
1614, 15syl 17 . . . . . . . . . 10 (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣𝐵) → ((𝐴𝑣) ∈ 𝒫 𝐴 ↔ (𝐴𝑣) ⊆ 𝐴))
175, 16mpbiri 248 . . . . . . . . 9 (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣𝐵) → (𝐴𝑣) ∈ 𝒫 𝐴)
18 simpl1 1084 . . . . . . . . . . . . 13 (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣𝐵) → 𝐵 ⊆ 𝒫 𝐴)
19 simpr 476 . . . . . . . . . . . . 13 (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣𝐵) → 𝑣𝐵)
2018, 19sseldd 3637 . . . . . . . . . . . 12 (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣𝐵) → 𝑣 ∈ 𝒫 𝐴)
2120elpwid 4203 . . . . . . . . . . 11 (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣𝐵) → 𝑣𝐴)
22 dfss4 3891 . . . . . . . . . . 11 (𝑣𝐴 ↔ (𝐴 ∖ (𝐴𝑣)) = 𝑣)
2321, 22sylib 208 . . . . . . . . . 10 (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣𝐵) → (𝐴 ∖ (𝐴𝑣)) = 𝑣)
2423, 19eqeltrd 2730 . . . . . . . . 9 (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣𝐵) → (𝐴 ∖ (𝐴𝑣)) ∈ 𝐵)
25 difeq2 3755 . . . . . . . . . . 11 (𝑐 = (𝐴𝑣) → (𝐴𝑐) = (𝐴 ∖ (𝐴𝑣)))
2625eleq1d 2715 . . . . . . . . . 10 (𝑐 = (𝐴𝑣) → ((𝐴𝑐) ∈ 𝐵 ↔ (𝐴 ∖ (𝐴𝑣)) ∈ 𝐵))
2726elrab 3396 . . . . . . . . 9 ((𝐴𝑣) ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ↔ ((𝐴𝑣) ∈ 𝒫 𝐴 ∧ (𝐴 ∖ (𝐴𝑣)) ∈ 𝐵))
2817, 24, 27sylanbrc 699 . . . . . . . 8 (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣𝐵) → (𝐴𝑣) ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵})
29 simpl3 1086 . . . . . . . 8 (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣𝐵) → ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝜑)
30 fin23lem11.2 . . . . . . . . . 10 (𝑤 = (𝐴𝑣) → (𝜑𝜃))
3130notbid 307 . . . . . . . . 9 (𝑤 = (𝐴𝑣) → (¬ 𝜑 ↔ ¬ 𝜃))
3231rspcva 3338 . . . . . . . 8 (((𝐴𝑣) ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝜑) → ¬ 𝜃)
3328, 29, 32syl2anc 694 . . . . . . 7 (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣𝐵) → ¬ 𝜃)
34 simplrl 817 . . . . . . . . . . 11 (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝐵)) ∧ 𝑣𝐵) → 𝑥 ∈ 𝒫 𝐴)
3534elpwid 4203 . . . . . . . . . 10 (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝐵)) ∧ 𝑣𝐵) → 𝑥𝐴)
36 ssel2 3631 . . . . . . . . . . . 12 ((𝐵 ⊆ 𝒫 𝐴𝑣𝐵) → 𝑣 ∈ 𝒫 𝐴)
3736adantlr 751 . . . . . . . . . . 11 (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝐵)) ∧ 𝑣𝐵) → 𝑣 ∈ 𝒫 𝐴)
3837elpwid 4203 . . . . . . . . . 10 (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝐵)) ∧ 𝑣𝐵) → 𝑣𝐴)
39 fin23lem11.3 . . . . . . . . . 10 ((𝑥𝐴𝑣𝐴) → (𝜒𝜃))
4035, 38, 39syl2anc 694 . . . . . . . . 9 (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝐵)) ∧ 𝑣𝐵) → (𝜒𝜃))
4140notbid 307 . . . . . . . 8 (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝐵)) ∧ 𝑣𝐵) → (¬ 𝜒 ↔ ¬ 𝜃))
42413adantl3 1239 . . . . . . 7 (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣𝐵) → (¬ 𝜒 ↔ ¬ 𝜃))
4333, 42mpbird 247 . . . . . 6 (((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝜑) ∧ 𝑣𝐵) → ¬ 𝜒)
4443ralrimiva 2995 . . . . 5 ((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝜑) → ∀𝑣𝐵 ¬ 𝜒)
45 fin23lem11.1 . . . . . . . 8 (𝑧 = (𝐴𝑥) → (𝜓𝜒))
4645notbid 307 . . . . . . 7 (𝑧 = (𝐴𝑥) → (¬ 𝜓 ↔ ¬ 𝜒))
4746ralbidv 3015 . . . . . 6 (𝑧 = (𝐴𝑥) → (∀𝑣𝐵 ¬ 𝜓 ↔ ∀𝑣𝐵 ¬ 𝜒))
4847rspcev 3340 . . . . 5 (((𝐴𝑥) ∈ 𝐵 ∧ ∀𝑣𝐵 ¬ 𝜒) → ∃𝑧𝐵𝑣𝐵 ¬ 𝜓)
494, 44, 48syl2anc 694 . . . 4 ((𝐵 ⊆ 𝒫 𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝐵) ∧ ∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝜑) → ∃𝑧𝐵𝑣𝐵 ¬ 𝜓)
50493exp 1283 . . 3 (𝐵 ⊆ 𝒫 𝐴 → ((𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝐵) → (∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝜑 → ∃𝑧𝐵𝑣𝐵 ¬ 𝜓)))
513, 50syl5bi 232 . 2 (𝐵 ⊆ 𝒫 𝐴 → (𝑥 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} → (∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝜑 → ∃𝑧𝐵𝑣𝐵 ¬ 𝜓)))
5251rexlimdv 3059 1 (𝐵 ⊆ 𝒫 𝐴 → (∃𝑥 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵}∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝜑 → ∃𝑧𝐵𝑣𝐵 ¬ 𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942  {crab 2945  Vcvv 3231   ∖ cdif 3604   ∪ cun 3605   ⊆ wss 3607  𝒫 cpw 4191 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-pw 4193  df-sn 4211  df-pr 4213  df-uni 4469 This theorem is referenced by:  fin2i2  9178  isfin2-2  9179
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