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Theorem mreexexlem4d 17021
Description: Induction step of the induction in mreexexd 17022. (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 484 . . 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 484 . . 3 ((𝜑𝐹 = ∅) → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))
7 mreexexlem2d.5 . . . 4 (𝜑𝐹 ⊆ (𝑋𝐻))
87adantr 484 . . 3 ((𝜑𝐹 = ∅) → 𝐹 ⊆ (𝑋𝐻))
9 mreexexlem2d.6 . . . 4 (𝜑𝐺 ⊆ (𝑋𝐻))
109adantr 484 . . 3 ((𝜑𝐹 = ∅) → 𝐺 ⊆ (𝑋𝐻))
11 mreexexlem2d.7 . . . 4 (𝜑𝐹 ⊆ (𝑁‘(𝐺𝐻)))
1211adantr 484 . . 3 ((𝜑𝐹 = ∅) → 𝐹 ⊆ (𝑁‘(𝐺𝐻)))
13 mreexexlem2d.8 . . . 4 (𝜑 → (𝐹𝐻) ∈ 𝐼)
1413adantr 484 . . 3 ((𝜑𝐹 = ∅) → (𝐹𝐻) ∈ 𝐼)
15 animorrl 980 . . 3 ((𝜑𝐹 = ∅) → (𝐹 = ∅ ∨ 𝐺 = ∅))
162, 3, 4, 6, 8, 10, 12, 14, 15mreexexlem3d 17020 . 2 ((𝜑𝐹 = ∅) → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))
17 n0 4235 . . . . 5 (𝐹 ≠ ∅ ↔ ∃𝑟 𝑟𝐹)
1817biimpi 219 . . . 4 (𝐹 ≠ ∅ → ∃𝑟 𝑟𝐹)
1918adantl 485 . . 3 ((𝜑𝐹 ≠ ∅) → ∃𝑟 𝑟𝐹)
201adantr 484 . . . . . 6 ((𝜑𝑟𝐹) → 𝐴 ∈ (Moore‘𝑋))
215adantr 484 . . . . . 6 ((𝜑𝑟𝐹) → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))
227adantr 484 . . . . . 6 ((𝜑𝑟𝐹) → 𝐹 ⊆ (𝑋𝐻))
239adantr 484 . . . . . 6 ((𝜑𝑟𝐹) → 𝐺 ⊆ (𝑋𝐻))
2411adantr 484 . . . . . 6 ((𝜑𝑟𝐹) → 𝐹 ⊆ (𝑁‘(𝐺𝐻)))
2513adantr 484 . . . . . 6 ((𝜑𝑟𝐹) → (𝐹𝐻) ∈ 𝐼)
26 simpr 488 . . . . . 6 ((𝜑𝑟𝐹) → 𝑟𝐹)
2720, 3, 4, 21, 22, 23, 24, 25, 26mreexexlem2d 17019 . . . . 5 ((𝜑𝑟𝐹) → ∃𝑞𝐺𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))
28 3anass 1096 . . . . . 6 ((𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼) ↔ (𝑞𝐺 ∧ (¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)))
291ad2antrr 726 . . . . . . . . 9 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐴 ∈ (Moore‘𝑋))
3029elfvexd 6708 . . . . . . . 8 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝑋 ∈ V)
31 simpr2 1196 . . . . . . . . . . 11 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}))
32 difsnb 4694 . . . . . . . . . . 11 𝑞 ∈ (𝐹 ∖ {𝑟}) ↔ ((𝐹 ∖ {𝑟}) ∖ {𝑞}) = (𝐹 ∖ {𝑟}))
3331, 32sylib 221 . . . . . . . . . 10 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ((𝐹 ∖ {𝑟}) ∖ {𝑞}) = (𝐹 ∖ {𝑟}))
347ad2antrr 726 . . . . . . . . . . . 12 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐹 ⊆ (𝑋𝐻))
3534ssdifssd 4033 . . . . . . . . . . 11 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ∖ {𝑟}) ⊆ (𝑋𝐻))
3635ssdifd 4031 . . . . . . . . . 10 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ((𝐹 ∖ {𝑟}) ∖ {𝑞}) ⊆ ((𝑋𝐻) ∖ {𝑞}))
3733, 36eqsstrrd 3916 . . . . . . . . 9 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ∖ {𝑟}) ⊆ ((𝑋𝐻) ∖ {𝑞}))
38 difun1 4180 . . . . . . . . 9 (𝑋 ∖ (𝐻 ∪ {𝑞})) = ((𝑋𝐻) ∖ {𝑞})
3937, 38sseqtrrdi 3928 . . . . . . . 8 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ∖ {𝑟}) ⊆ (𝑋 ∖ (𝐻 ∪ {𝑞})))
409ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐺 ⊆ (𝑋𝐻))
4140ssdifd 4031 . . . . . . . . 9 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐺 ∖ {𝑞}) ⊆ ((𝑋𝐻) ∖ {𝑞}))
4241, 38sseqtrrdi 3928 . . . . . . . 8 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐺 ∖ {𝑞}) ⊆ (𝑋 ∖ (𝐻 ∪ {𝑞})))
4311ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐹 ⊆ (𝑁‘(𝐺𝐻)))
44 simpr1 1195 . . . . . . . . . . . 12 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝑞𝐺)
45 uncom 4043 . . . . . . . . . . . . . 14 (𝐻 ∪ {𝑞}) = ({𝑞} ∪ 𝐻)
4645uneq2i 4050 . . . . . . . . . . . . 13 ((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞})) = ((𝐺 ∖ {𝑞}) ∪ ({𝑞} ∪ 𝐻))
47 unass 4056 . . . . . . . . . . . . . 14 (((𝐺 ∖ {𝑞}) ∪ {𝑞}) ∪ 𝐻) = ((𝐺 ∖ {𝑞}) ∪ ({𝑞} ∪ 𝐻))
48 difsnid 4698 . . . . . . . . . . . . . . 15 (𝑞𝐺 → ((𝐺 ∖ {𝑞}) ∪ {𝑞}) = 𝐺)
4948uneq1d 4052 . . . . . . . . . . . . . 14 (𝑞𝐺 → (((𝐺 ∖ {𝑞}) ∪ {𝑞}) ∪ 𝐻) = (𝐺𝐻))
5047, 49eqtr3id 2787 . . . . . . . . . . . . 13 (𝑞𝐺 → ((𝐺 ∖ {𝑞}) ∪ ({𝑞} ∪ 𝐻)) = (𝐺𝐻))
5146, 50syl5eq 2785 . . . . . . . . . . . 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 6678 . . . . . . . . . 10 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝑁‘((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞}))) = (𝑁‘(𝐺𝐻)))
5443, 53sseqtrrd 3918 . . . . . . . . 9 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐹 ⊆ (𝑁‘((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞}))))
5554ssdifssd 4033 . . . . . . . 8 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ∖ {𝑟}) ⊆ (𝑁‘((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞}))))
56 simpr3 1197 . . . . . . . 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 726 . . . . . . . . 9 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ≈ suc 𝐿𝐺 ≈ suc 𝐿))
59 mreexexlem4d.9 . . . . . . . . . . . 12 (𝜑𝐿 ∈ ω)
6059ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐿 ∈ ω)
61 simplr 769 . . . . . . . . . . 11 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝑟𝐹)
62 3anan12 1097 . . . . . . . . . . . . 13 ((𝐿 ∈ ω ∧ 𝐹 ≈ suc 𝐿𝑟𝐹) ↔ (𝐹 ≈ suc 𝐿 ∧ (𝐿 ∈ ω ∧ 𝑟𝐹)))
63 dif1en 8761 . . . . . . . . . . . . 13 ((𝐿 ∈ ω ∧ 𝐹 ≈ suc 𝐿𝑟𝐹) → (𝐹 ∖ {𝑟}) ≈ 𝐿)
6462, 63sylbir 238 . . . . . . . . . . . 12 ((𝐹 ≈ suc 𝐿 ∧ (𝐿 ∈ ω ∧ 𝑟𝐹)) → (𝐹 ∖ {𝑟}) ≈ 𝐿)
6564expcom 417 . . . . . . . . . . 11 ((𝐿 ∈ ω ∧ 𝑟𝐹) → (𝐹 ≈ suc 𝐿 → (𝐹 ∖ {𝑟}) ≈ 𝐿))
6660, 61, 65syl2anc 587 . . . . . . . . . 10 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ≈ suc 𝐿 → (𝐹 ∖ {𝑟}) ≈ 𝐿))
67 3anan12 1097 . . . . . . . . . . . . 13 ((𝐿 ∈ ω ∧ 𝐺 ≈ suc 𝐿𝑞𝐺) ↔ (𝐺 ≈ suc 𝐿 ∧ (𝐿 ∈ ω ∧ 𝑞𝐺)))
68 dif1en 8761 . . . . . . . . . . . . 13 ((𝐿 ∈ ω ∧ 𝐺 ≈ suc 𝐿𝑞𝐺) → (𝐺 ∖ {𝑞}) ≈ 𝐿)
6967, 68sylbir 238 . . . . . . . . . . . 12 ((𝐺 ≈ suc 𝐿 ∧ (𝐿 ∈ ω ∧ 𝑞𝐺)) → (𝐺 ∖ {𝑞}) ≈ 𝐿)
7069expcom 417 . . . . . . . . . . 11 ((𝐿 ∈ ω ∧ 𝑞𝐺) → (𝐺 ≈ suc 𝐿 → (𝐺 ∖ {𝑞}) ≈ 𝐿))
7160, 44, 70syl2anc 587 . . . . . . . . . 10 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐺 ≈ suc 𝐿 → (𝐺 ∖ {𝑞}) ≈ 𝐿))
7266, 71orim12d 964 . . . . . . . . 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 726 . . . . . . . 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 17018 . . . . . . 7 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ∃𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞})((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))
7730adantr 484 . . . . . . . . . 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 730 . . . . . . . . . . 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 4025 . . . . . . . . . 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 5192 . . . . . . . . 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 771 . . . . . . . . . . . 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 4499 . . . . . . . . . . 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 4025 . . . . . . . . . 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 1216 . . . . . . . . . . 11 ((((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝑞𝐺)
8584snssd 4697 . . . . . . . . . 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 4076 . . . . . . . . 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 5194 . . . . . . . 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 4698 . . . . . . . . . 10 (𝑟𝐹 → ((𝐹 ∖ {𝑟}) ∪ {𝑟}) = 𝐹)
8988ad3antlr 731 . . . . . . . . 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 781 . . . . . . . . . 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 8640 . . . . . . . . . . . 12 ((𝑟 ∈ V ∧ 𝑞 ∈ V) → {𝑟} ≈ {𝑞})
9291el2v 3406 . . . . . . . . . . 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 disjdifr 4362 . . . . . . . . . . 11 ((𝐹 ∖ {𝑟}) ∩ {𝑟}) = ∅
9594a1i 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}) = ∅)
96 ssdifin0 4372 . . . . . . . . . . 11 (𝑖 ⊆ (𝐺 ∖ {𝑞}) → (𝑖 ∩ {𝑞}) = ∅)
9782, 96syl 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}) = ∅)
98 unen 8644 . . . . . . . . . 10 ((((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ {𝑟} ≈ {𝑞}) ∧ (((𝐹 ∖ {𝑟}) ∩ {𝑟}) = ∅ ∧ (𝑖 ∩ {𝑞}) = ∅)) → ((𝐹 ∖ {𝑟}) ∪ {𝑟}) ≈ (𝑖 ∪ {𝑞}))
9990, 93, 95, 97, 98syl22anc 838 . . . . . . . . 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}))
10089, 99eqbrtrrd 5054 . . . . . . . 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}))
101 unass 4056 . . . . . . . . . 10 ((𝑖 ∪ {𝑞}) ∪ 𝐻) = (𝑖 ∪ ({𝑞} ∪ 𝐻))
102 uncom 4043 . . . . . . . . . . 11 ({𝑞} ∪ 𝐻) = (𝐻 ∪ {𝑞})
103102uneq2i 4050 . . . . . . . . . 10 (𝑖 ∪ ({𝑞} ∪ 𝐻)) = (𝑖 ∪ (𝐻 ∪ {𝑞}))
104101, 103eqtr2i 2762 . . . . . . . . 9 (𝑖 ∪ (𝐻 ∪ {𝑞})) = ((𝑖 ∪ {𝑞}) ∪ 𝐻)
105 simprrr 782 . . . . . . . . 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)
106104, 105eqeltrrid 2838 . . . . . . . 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)
107 breq2 5034 . . . . . . . . . 10 (𝑗 = (𝑖 ∪ {𝑞}) → (𝐹𝑗𝐹 ≈ (𝑖 ∪ {𝑞})))
108 uneq1 4046 . . . . . . . . . . 11 (𝑗 = (𝑖 ∪ {𝑞}) → (𝑗𝐻) = ((𝑖 ∪ {𝑞}) ∪ 𝐻))
109108eleq1d 2817 . . . . . . . . . 10 (𝑗 = (𝑖 ∪ {𝑞}) → ((𝑗𝐻) ∈ 𝐼 ↔ ((𝑖 ∪ {𝑞}) ∪ 𝐻) ∈ 𝐼))
110107, 109anbi12d 634 . . . . . . . . 9 (𝑗 = (𝑖 ∪ {𝑞}) → ((𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼) ↔ (𝐹 ≈ (𝑖 ∪ {𝑞}) ∧ ((𝑖 ∪ {𝑞}) ∪ 𝐻) ∈ 𝐼)))
111110rspcev 3526 . . . . . . . 8 (((𝑖 ∪ {𝑞}) ∈ 𝒫 𝐺 ∧ (𝐹 ≈ (𝑖 ∪ {𝑞}) ∧ ((𝑖 ∪ {𝑞}) ∪ 𝐻) ∈ 𝐼)) → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))
11287, 100, 106, 111syl12anc 836 . . . . . . 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))
11376, 112rexlimddv 3201 . . . . . 6 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))
11428, 113sylan2br 598 . . . . 5 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ (¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))
11527, 114rexlimddv 3201 . . . 4 ((𝜑𝑟𝐹) → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))
116115adantlr 715 . . 3 (((𝜑𝐹 ≠ ∅) ∧ 𝑟𝐹) → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))
11719, 116exlimddv 1942 . 2 ((𝜑𝐹 ≠ ∅) → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))
11816, 117pm2.61dane 3021 1 (𝜑 → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wo 846  w3a 1088  wal 1540   = wceq 1542  wex 1786  wcel 2114  wne 2934  wral 3053  wrex 3054  Vcvv 3398  cdif 3840  cun 3841  cin 3842  wss 3843  c0 4211  𝒫 cpw 4488  {csn 4516   class class class wbr 5030  suc csuc 6174  cfv 6339  ωcom 7599  cen 8552  Moorecmre 16956  mrClscmrc 16957  mrIndcmri 16958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-int 4837  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ord 6175  df-on 6176  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-om 7600  df-en 8556  df-mre 16960  df-mrc 16961  df-mri 16962
This theorem is referenced by:  mreexexd  17022
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