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Theorem mreexexlem4d 16917
 Description: Induction step of the induction in mreexexd 16918. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
mreexexlem2d.1 (𝜑𝐴 ∈ (Moore‘𝑋))
mreexexlem2d.2 𝑁 = (mrCls‘𝐴)
mreexexlem2d.3 𝐼 = (mrInd‘𝐴)
mreexexlem2d.4 (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))
mreexexlem2d.5 (𝜑𝐹 ⊆ (𝑋𝐻))
mreexexlem2d.6 (𝜑𝐺 ⊆ (𝑋𝐻))
mreexexlem2d.7 (𝜑𝐹 ⊆ (𝑁‘(𝐺𝐻)))
mreexexlem2d.8 (𝜑 → (𝐹𝐻) ∈ 𝐼)
mreexexlem4d.9 (𝜑𝐿 ∈ ω)
mreexexlem4d.A (𝜑 → ∀𝑓 ∈ 𝒫 (𝑋)∀𝑔 ∈ 𝒫 (𝑋)(((𝑓𝐿𝑔𝐿) ∧ 𝑓 ⊆ (𝑁‘(𝑔)) ∧ (𝑓) ∈ 𝐼) → ∃𝑗 ∈ 𝒫 𝑔(𝑓𝑗 ∧ (𝑗) ∈ 𝐼)))
mreexexlem4d.B (𝜑 → (𝐹 ≈ suc 𝐿𝐺 ≈ suc 𝐿))
Assertion
Ref Expression
mreexexlem4d (𝜑 → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))
Distinct variable groups:   𝑓,𝑔,,𝑋   𝑓,𝐼,𝑗,𝑔,   𝑓,𝐿,𝑔,   𝑓,𝑁,𝑔,   𝑦,𝑠,𝑧,𝑁   𝐹,𝑠,𝑦,𝑧   𝐺,𝑠,𝑦,𝑧   𝐻,𝑠,𝑦,𝑧   𝜑,𝑠,𝑦,𝑧   𝑗,𝐹   𝑗,𝐺   𝑗,𝐻   𝑋,𝑠,𝑦
Allowed substitution hints:   𝜑(𝑓,𝑔,,𝑗)   𝐴(𝑦,𝑧,𝑓,𝑔,,𝑗,𝑠)   𝐹(𝑓,𝑔,)   𝐺(𝑓,𝑔,)   𝐻(𝑓,𝑔,)   𝐼(𝑦,𝑧,𝑠)   𝐿(𝑦,𝑧,𝑗,𝑠)   𝑁(𝑗)   𝑋(𝑧,𝑗)

Proof of Theorem mreexexlem4d
Dummy variables 𝑖 𝑞 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mreexexlem2d.1 . . . 4 (𝜑𝐴 ∈ (Moore‘𝑋))
21adantr 483 . . 3 ((𝜑𝐹 = ∅) → 𝐴 ∈ (Moore‘𝑋))
3 mreexexlem2d.2 . . 3 𝑁 = (mrCls‘𝐴)
4 mreexexlem2d.3 . . 3 𝐼 = (mrInd‘𝐴)
5 mreexexlem2d.4 . . . 4 (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))
65adantr 483 . . 3 ((𝜑𝐹 = ∅) → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))
7 mreexexlem2d.5 . . . 4 (𝜑𝐹 ⊆ (𝑋𝐻))
87adantr 483 . . 3 ((𝜑𝐹 = ∅) → 𝐹 ⊆ (𝑋𝐻))
9 mreexexlem2d.6 . . . 4 (𝜑𝐺 ⊆ (𝑋𝐻))
109adantr 483 . . 3 ((𝜑𝐹 = ∅) → 𝐺 ⊆ (𝑋𝐻))
11 mreexexlem2d.7 . . . 4 (𝜑𝐹 ⊆ (𝑁‘(𝐺𝐻)))
1211adantr 483 . . 3 ((𝜑𝐹 = ∅) → 𝐹 ⊆ (𝑁‘(𝐺𝐻)))
13 mreexexlem2d.8 . . . 4 (𝜑 → (𝐹𝐻) ∈ 𝐼)
1413adantr 483 . . 3 ((𝜑𝐹 = ∅) → (𝐹𝐻) ∈ 𝐼)
15 animorrl 977 . . 3 ((𝜑𝐹 = ∅) → (𝐹 = ∅ ∨ 𝐺 = ∅))
162, 3, 4, 6, 8, 10, 12, 14, 15mreexexlem3d 16916 . 2 ((𝜑𝐹 = ∅) → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))
17 n0 4309 . . . . 5 (𝐹 ≠ ∅ ↔ ∃𝑟 𝑟𝐹)
1817biimpi 218 . . . 4 (𝐹 ≠ ∅ → ∃𝑟 𝑟𝐹)
1918adantl 484 . . 3 ((𝜑𝐹 ≠ ∅) → ∃𝑟 𝑟𝐹)
201adantr 483 . . . . . 6 ((𝜑𝑟𝐹) → 𝐴 ∈ (Moore‘𝑋))
215adantr 483 . . . . . 6 ((𝜑𝑟𝐹) → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))
227adantr 483 . . . . . 6 ((𝜑𝑟𝐹) → 𝐹 ⊆ (𝑋𝐻))
239adantr 483 . . . . . 6 ((𝜑𝑟𝐹) → 𝐺 ⊆ (𝑋𝐻))
2411adantr 483 . . . . . 6 ((𝜑𝑟𝐹) → 𝐹 ⊆ (𝑁‘(𝐺𝐻)))
2513adantr 483 . . . . . 6 ((𝜑𝑟𝐹) → (𝐹𝐻) ∈ 𝐼)
26 simpr 487 . . . . . 6 ((𝜑𝑟𝐹) → 𝑟𝐹)
2720, 3, 4, 21, 22, 23, 24, 25, 26mreexexlem2d 16915 . . . . 5 ((𝜑𝑟𝐹) → ∃𝑞𝐺𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))
28 3anass 1091 . . . . . 6 ((𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼) ↔ (𝑞𝐺 ∧ (¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)))
291ad2antrr 724 . . . . . . . . 9 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐴 ∈ (Moore‘𝑋))
3029elfvexd 6703 . . . . . . . 8 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝑋 ∈ V)
31 simpr2 1191 . . . . . . . . . . 11 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}))
32 difsnb 4738 . . . . . . . . . . 11 𝑞 ∈ (𝐹 ∖ {𝑟}) ↔ ((𝐹 ∖ {𝑟}) ∖ {𝑞}) = (𝐹 ∖ {𝑟}))
3331, 32sylib 220 . . . . . . . . . 10 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ((𝐹 ∖ {𝑟}) ∖ {𝑞}) = (𝐹 ∖ {𝑟}))
347ad2antrr 724 . . . . . . . . . . . 12 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐹 ⊆ (𝑋𝐻))
3534ssdifssd 4118 . . . . . . . . . . 11 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ∖ {𝑟}) ⊆ (𝑋𝐻))
3635ssdifd 4116 . . . . . . . . . 10 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ((𝐹 ∖ {𝑟}) ∖ {𝑞}) ⊆ ((𝑋𝐻) ∖ {𝑞}))
3733, 36eqsstrrd 4005 . . . . . . . . 9 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ∖ {𝑟}) ⊆ ((𝑋𝐻) ∖ {𝑞}))
38 difun1 4263 . . . . . . . . 9 (𝑋 ∖ (𝐻 ∪ {𝑞})) = ((𝑋𝐻) ∖ {𝑞})
3937, 38sseqtrrdi 4017 . . . . . . . 8 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ∖ {𝑟}) ⊆ (𝑋 ∖ (𝐻 ∪ {𝑞})))
409ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐺 ⊆ (𝑋𝐻))
4140ssdifd 4116 . . . . . . . . 9 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐺 ∖ {𝑞}) ⊆ ((𝑋𝐻) ∖ {𝑞}))
4241, 38sseqtrrdi 4017 . . . . . . . 8 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐺 ∖ {𝑞}) ⊆ (𝑋 ∖ (𝐻 ∪ {𝑞})))
4311ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐹 ⊆ (𝑁‘(𝐺𝐻)))
44 simpr1 1190 . . . . . . . . . . . 12 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝑞𝐺)
45 uncom 4128 . . . . . . . . . . . . . 14 (𝐻 ∪ {𝑞}) = ({𝑞} ∪ 𝐻)
4645uneq2i 4135 . . . . . . . . . . . . 13 ((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞})) = ((𝐺 ∖ {𝑞}) ∪ ({𝑞} ∪ 𝐻))
47 unass 4141 . . . . . . . . . . . . . 14 (((𝐺 ∖ {𝑞}) ∪ {𝑞}) ∪ 𝐻) = ((𝐺 ∖ {𝑞}) ∪ ({𝑞} ∪ 𝐻))
48 difsnid 4742 . . . . . . . . . . . . . . 15 (𝑞𝐺 → ((𝐺 ∖ {𝑞}) ∪ {𝑞}) = 𝐺)
4948uneq1d 4137 . . . . . . . . . . . . . 14 (𝑞𝐺 → (((𝐺 ∖ {𝑞}) ∪ {𝑞}) ∪ 𝐻) = (𝐺𝐻))
5047, 49syl5eqr 2870 . . . . . . . . . . . . 13 (𝑞𝐺 → ((𝐺 ∖ {𝑞}) ∪ ({𝑞} ∪ 𝐻)) = (𝐺𝐻))
5146, 50syl5eq 2868 . . . . . . . . . . . 12 (𝑞𝐺 → ((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞})) = (𝐺𝐻))
5244, 51syl 17 . . . . . . . . . . 11 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞})) = (𝐺𝐻))
5352fveq2d 6673 . . . . . . . . . 10 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝑁‘((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞}))) = (𝑁‘(𝐺𝐻)))
5443, 53sseqtrrd 4007 . . . . . . . . 9 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐹 ⊆ (𝑁‘((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞}))))
5554ssdifssd 4118 . . . . . . . 8 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ∖ {𝑟}) ⊆ (𝑁‘((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞}))))
56 simpr3 1192 . . . . . . . 8 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)
57 mreexexlem4d.B . . . . . . . . . 10 (𝜑 → (𝐹 ≈ suc 𝐿𝐺 ≈ suc 𝐿))
5857ad2antrr 724 . . . . . . . . 9 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ≈ suc 𝐿𝐺 ≈ suc 𝐿))
59 mreexexlem4d.9 . . . . . . . . . . . 12 (𝜑𝐿 ∈ ω)
6059ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐿 ∈ ω)
61 simplr 767 . . . . . . . . . . 11 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝑟𝐹)
62 3anan12 1092 . . . . . . . . . . . . 13 ((𝐿 ∈ ω ∧ 𝐹 ≈ suc 𝐿𝑟𝐹) ↔ (𝐹 ≈ suc 𝐿 ∧ (𝐿 ∈ ω ∧ 𝑟𝐹)))
63 dif1en 8750 . . . . . . . . . . . . 13 ((𝐿 ∈ ω ∧ 𝐹 ≈ suc 𝐿𝑟𝐹) → (𝐹 ∖ {𝑟}) ≈ 𝐿)
6462, 63sylbir 237 . . . . . . . . . . . 12 ((𝐹 ≈ suc 𝐿 ∧ (𝐿 ∈ ω ∧ 𝑟𝐹)) → (𝐹 ∖ {𝑟}) ≈ 𝐿)
6564expcom 416 . . . . . . . . . . 11 ((𝐿 ∈ ω ∧ 𝑟𝐹) → (𝐹 ≈ suc 𝐿 → (𝐹 ∖ {𝑟}) ≈ 𝐿))
6660, 61, 65syl2anc 586 . . . . . . . . . 10 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ≈ suc 𝐿 → (𝐹 ∖ {𝑟}) ≈ 𝐿))
67 3anan12 1092 . . . . . . . . . . . . 13 ((𝐿 ∈ ω ∧ 𝐺 ≈ suc 𝐿𝑞𝐺) ↔ (𝐺 ≈ suc 𝐿 ∧ (𝐿 ∈ ω ∧ 𝑞𝐺)))
68 dif1en 8750 . . . . . . . . . . . . 13 ((𝐿 ∈ ω ∧ 𝐺 ≈ suc 𝐿𝑞𝐺) → (𝐺 ∖ {𝑞}) ≈ 𝐿)
6967, 68sylbir 237 . . . . . . . . . . . 12 ((𝐺 ≈ suc 𝐿 ∧ (𝐿 ∈ ω ∧ 𝑞𝐺)) → (𝐺 ∖ {𝑞}) ≈ 𝐿)
7069expcom 416 . . . . . . . . . . 11 ((𝐿 ∈ ω ∧ 𝑞𝐺) → (𝐺 ≈ suc 𝐿 → (𝐺 ∖ {𝑞}) ≈ 𝐿))
7160, 44, 70syl2anc 586 . . . . . . . . . 10 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐺 ≈ suc 𝐿 → (𝐺 ∖ {𝑞}) ≈ 𝐿))
7266, 71orim12d 961 . . . . . . . . 9 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ((𝐹 ≈ suc 𝐿𝐺 ≈ suc 𝐿) → ((𝐹 ∖ {𝑟}) ≈ 𝐿 ∨ (𝐺 ∖ {𝑞}) ≈ 𝐿)))
7358, 72mpd 15 . . . . . . . 8 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ((𝐹 ∖ {𝑟}) ≈ 𝐿 ∨ (𝐺 ∖ {𝑞}) ≈ 𝐿))
74 mreexexlem4d.A . . . . . . . . 9 (𝜑 → ∀𝑓 ∈ 𝒫 (𝑋)∀𝑔 ∈ 𝒫 (𝑋)(((𝑓𝐿𝑔𝐿) ∧ 𝑓 ⊆ (𝑁‘(𝑔)) ∧ (𝑓) ∈ 𝐼) → ∃𝑗 ∈ 𝒫 𝑔(𝑓𝑗 ∧ (𝑗) ∈ 𝐼)))
7574ad2antrr 724 . . . . . . . 8 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ∀𝑓 ∈ 𝒫 (𝑋)∀𝑔 ∈ 𝒫 (𝑋)(((𝑓𝐿𝑔𝐿) ∧ 𝑓 ⊆ (𝑁‘(𝑔)) ∧ (𝑓) ∈ 𝐼) → ∃𝑗 ∈ 𝒫 𝑔(𝑓𝑗 ∧ (𝑗) ∈ 𝐼)))
7630, 39, 42, 55, 56, 73, 75mreexexlemd 16914 . . . . . . 7 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ∃𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞})((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))
7730adantr 483 . . . . . . . . . 10 ((((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝑋 ∈ V)
789ad3antrrr 728 . . . . . . . . . . 11 ((((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝐺 ⊆ (𝑋𝐻))
7978difss2d 4110 . . . . . . . . . 10 ((((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝐺𝑋)
8077, 79ssexd 5227 . . . . . . . . 9 ((((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝐺 ∈ V)
81 simprl 769 . . . . . . . . . . . 12 ((((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}))
8281elpwid 4549 . . . . . . . . . . 11 ((((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝑖 ⊆ (𝐺 ∖ {𝑞}))
8382difss2d 4110 . . . . . . . . . 10 ((((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝑖𝐺)
84 simplr1 1211 . . . . . . . . . . 11 ((((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝑞𝐺)
8584snssd 4741 . . . . . . . . . 10 ((((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → {𝑞} ⊆ 𝐺)
8683, 85unssd 4161 . . . . . . . . 9 ((((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → (𝑖 ∪ {𝑞}) ⊆ 𝐺)
8780, 86sselpwd 5229 . . . . . . . 8 ((((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → (𝑖 ∪ {𝑞}) ∈ 𝒫 𝐺)
88 difsnid 4742 . . . . . . . . . 10 (𝑟𝐹 → ((𝐹 ∖ {𝑟}) ∪ {𝑟}) = 𝐹)
8988ad3antlr 729 . . . . . . . . 9 ((((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → ((𝐹 ∖ {𝑟}) ∪ {𝑟}) = 𝐹)
90 simprrl 779 . . . . . . . . . 10 ((((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → (𝐹 ∖ {𝑟}) ≈ 𝑖)
91 en2sn 8592 . . . . . . . . . . . 12 ((𝑟 ∈ V ∧ 𝑞 ∈ V) → {𝑟} ≈ {𝑞})
9291el2v 3501 . . . . . . . . . . 11 {𝑟} ≈ {𝑞}
9392a1i 11 . . . . . . . . . 10 ((((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → {𝑟} ≈ {𝑞})
94 incom 4177 . . . . . . . . . . . 12 ((𝐹 ∖ {𝑟}) ∩ {𝑟}) = ({𝑟} ∩ (𝐹 ∖ {𝑟}))
95 disjdif 4420 . . . . . . . . . . . 12 ({𝑟} ∩ (𝐹 ∖ {𝑟})) = ∅
9694, 95eqtri 2844 . . . . . . . . . . 11 ((𝐹 ∖ {𝑟}) ∩ {𝑟}) = ∅
9796a1i 11 . . . . . . . . . 10 ((((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → ((𝐹 ∖ {𝑟}) ∩ {𝑟}) = ∅)
98 ssdifin0 4430 . . . . . . . . . . 11 (𝑖 ⊆ (𝐺 ∖ {𝑞}) → (𝑖 ∩ {𝑞}) = ∅)
9982, 98syl 17 . . . . . . . . . 10 ((((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → (𝑖 ∩ {𝑞}) = ∅)
100 unen 8595 . . . . . . . . . 10 ((((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ {𝑟} ≈ {𝑞}) ∧ (((𝐹 ∖ {𝑟}) ∩ {𝑟}) = ∅ ∧ (𝑖 ∩ {𝑞}) = ∅)) → ((𝐹 ∖ {𝑟}) ∪ {𝑟}) ≈ (𝑖 ∪ {𝑞}))
10190, 93, 97, 99, 100syl22anc 836 . . . . . . . . 9 ((((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → ((𝐹 ∖ {𝑟}) ∪ {𝑟}) ≈ (𝑖 ∪ {𝑞}))
10289, 101eqbrtrrd 5089 . . . . . . . 8 ((((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝐹 ≈ (𝑖 ∪ {𝑞}))
103 unass 4141 . . . . . . . . . 10 ((𝑖 ∪ {𝑞}) ∪ 𝐻) = (𝑖 ∪ ({𝑞} ∪ 𝐻))
104 uncom 4128 . . . . . . . . . . 11 ({𝑞} ∪ 𝐻) = (𝐻 ∪ {𝑞})
105104uneq2i 4135 . . . . . . . . . 10 (𝑖 ∪ ({𝑞} ∪ 𝐻)) = (𝑖 ∪ (𝐻 ∪ {𝑞}))
106103, 105eqtr2i 2845 . . . . . . . . 9 (𝑖 ∪ (𝐻 ∪ {𝑞})) = ((𝑖 ∪ {𝑞}) ∪ 𝐻)
107 simprrr 780 . . . . . . . . 9 ((((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)
108106, 107eqeltrrid 2918 . . . . . . . 8 ((((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → ((𝑖 ∪ {𝑞}) ∪ 𝐻) ∈ 𝐼)
109 breq2 5069 . . . . . . . . . 10 (𝑗 = (𝑖 ∪ {𝑞}) → (𝐹𝑗𝐹 ≈ (𝑖 ∪ {𝑞})))
110 uneq1 4131 . . . . . . . . . . 11 (𝑗 = (𝑖 ∪ {𝑞}) → (𝑗𝐻) = ((𝑖 ∪ {𝑞}) ∪ 𝐻))
111110eleq1d 2897 . . . . . . . . . 10 (𝑗 = (𝑖 ∪ {𝑞}) → ((𝑗𝐻) ∈ 𝐼 ↔ ((𝑖 ∪ {𝑞}) ∪ 𝐻) ∈ 𝐼))
112109, 111anbi12d 632 . . . . . . . . 9 (𝑗 = (𝑖 ∪ {𝑞}) → ((𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼) ↔ (𝐹 ≈ (𝑖 ∪ {𝑞}) ∧ ((𝑖 ∪ {𝑞}) ∪ 𝐻) ∈ 𝐼)))
113112rspcev 3622 . . . . . . . 8 (((𝑖 ∪ {𝑞}) ∈ 𝒫 𝐺 ∧ (𝐹 ≈ (𝑖 ∪ {𝑞}) ∧ ((𝑖 ∪ {𝑞}) ∪ 𝐻) ∈ 𝐼)) → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))
11487, 102, 108, 113syl12anc 834 . . . . . . 7 ((((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))
11576, 114rexlimddv 3291 . . . . . 6 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))
11628, 115sylan2br 596 . . . . 5 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ (¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))
11727, 116rexlimddv 3291 . . . 4 ((𝜑𝑟𝐹) → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))
118117adantlr 713 . . 3 (((𝜑𝐹 ≠ ∅) ∧ 𝑟𝐹) → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))
11919, 118exlimddv 1932 . 2 ((𝜑𝐹 ≠ ∅) → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))
12016, 119pm2.61dane 3104 1 (𝜑 → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∨ wo 843   ∧ w3a 1083  ∀wal 1531   = wceq 1533  ∃wex 1776   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  Vcvv 3494   ∖ cdif 3932   ∪ cun 3933   ∩ cin 3934   ⊆ wss 3935  ∅c0 4290  𝒫 cpw 4538  {csn 4566   class class class wbr 5065  suc csuc 6192  ‘cfv 6354  ωcom 7579   ≈ cen 8505  Moorecmre 16852  mrClscmrc 16853  mrIndcmri 16854 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-om 7580  df-1o 8101  df-er 8288  df-en 8509  df-fin 8512  df-mre 16856  df-mrc 16857  df-mri 16858 This theorem is referenced by:  mreexexd  16918
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