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Theorem mreexexlem4d 16239
Description: Induction step of the induction in mreexexd 16240. (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 481 . . 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 481 . . 3 ((𝜑𝐹 = ∅) → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))
7 mreexexlem2d.5 . . . 4 (𝜑𝐹 ⊆ (𝑋𝐻))
87adantr 481 . . 3 ((𝜑𝐹 = ∅) → 𝐹 ⊆ (𝑋𝐻))
9 mreexexlem2d.6 . . . 4 (𝜑𝐺 ⊆ (𝑋𝐻))
109adantr 481 . . 3 ((𝜑𝐹 = ∅) → 𝐺 ⊆ (𝑋𝐻))
11 mreexexlem2d.7 . . . 4 (𝜑𝐹 ⊆ (𝑁‘(𝐺𝐻)))
1211adantr 481 . . 3 ((𝜑𝐹 = ∅) → 𝐹 ⊆ (𝑁‘(𝐺𝐻)))
13 mreexexlem2d.8 . . . 4 (𝜑 → (𝐹𝐻) ∈ 𝐼)
1413adantr 481 . . 3 ((𝜑𝐹 = ∅) → (𝐹𝐻) ∈ 𝐼)
15 simpr 477 . . . 4 ((𝜑𝐹 = ∅) → 𝐹 = ∅)
1615orcd 407 . . 3 ((𝜑𝐹 = ∅) → (𝐹 = ∅ ∨ 𝐺 = ∅))
172, 3, 4, 6, 8, 10, 12, 14, 16mreexexlem3d 16238 . 2 ((𝜑𝐹 = ∅) → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))
18 n0 3912 . . . . 5 (𝐹 ≠ ∅ ↔ ∃𝑟 𝑟𝐹)
1918biimpi 206 . . . 4 (𝐹 ≠ ∅ → ∃𝑟 𝑟𝐹)
2019adantl 482 . . 3 ((𝜑𝐹 ≠ ∅) → ∃𝑟 𝑟𝐹)
211adantr 481 . . . . . 6 ((𝜑𝑟𝐹) → 𝐴 ∈ (Moore‘𝑋))
225adantr 481 . . . . . 6 ((𝜑𝑟𝐹) → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))
237adantr 481 . . . . . 6 ((𝜑𝑟𝐹) → 𝐹 ⊆ (𝑋𝐻))
249adantr 481 . . . . . 6 ((𝜑𝑟𝐹) → 𝐺 ⊆ (𝑋𝐻))
2511adantr 481 . . . . . 6 ((𝜑𝑟𝐹) → 𝐹 ⊆ (𝑁‘(𝐺𝐻)))
2613adantr 481 . . . . . 6 ((𝜑𝑟𝐹) → (𝐹𝐻) ∈ 𝐼)
27 simpr 477 . . . . . 6 ((𝜑𝑟𝐹) → 𝑟𝐹)
2821, 3, 4, 22, 23, 24, 25, 26, 27mreexexlem2d 16237 . . . . 5 ((𝜑𝑟𝐹) → ∃𝑞𝐺𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))
29 3anass 1040 . . . . . 6 ((𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼) ↔ (𝑞𝐺 ∧ (¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)))
301ad2antrr 761 . . . . . . . . 9 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐴 ∈ (Moore‘𝑋))
3130elfvexd 6184 . . . . . . . 8 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝑋 ∈ V)
32 simpr2 1066 . . . . . . . . . . 11 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}))
33 difsnb 4311 . . . . . . . . . . 11 𝑞 ∈ (𝐹 ∖ {𝑟}) ↔ ((𝐹 ∖ {𝑟}) ∖ {𝑞}) = (𝐹 ∖ {𝑟}))
3432, 33sylib 208 . . . . . . . . . 10 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ((𝐹 ∖ {𝑟}) ∖ {𝑞}) = (𝐹 ∖ {𝑟}))
357ad2antrr 761 . . . . . . . . . . . 12 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐹 ⊆ (𝑋𝐻))
3635ssdifssd 3731 . . . . . . . . . . 11 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ∖ {𝑟}) ⊆ (𝑋𝐻))
3736ssdifd 3729 . . . . . . . . . 10 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ((𝐹 ∖ {𝑟}) ∖ {𝑞}) ⊆ ((𝑋𝐻) ∖ {𝑞}))
3834, 37eqsstr3d 3624 . . . . . . . . 9 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ∖ {𝑟}) ⊆ ((𝑋𝐻) ∖ {𝑞}))
39 difun1 3868 . . . . . . . . 9 (𝑋 ∖ (𝐻 ∪ {𝑞})) = ((𝑋𝐻) ∖ {𝑞})
4038, 39syl6sseqr 3636 . . . . . . . 8 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ∖ {𝑟}) ⊆ (𝑋 ∖ (𝐻 ∪ {𝑞})))
419ad2antrr 761 . . . . . . . . . 10 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐺 ⊆ (𝑋𝐻))
4241ssdifd 3729 . . . . . . . . 9 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐺 ∖ {𝑞}) ⊆ ((𝑋𝐻) ∖ {𝑞}))
4342, 39syl6sseqr 3636 . . . . . . . 8 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐺 ∖ {𝑞}) ⊆ (𝑋 ∖ (𝐻 ∪ {𝑞})))
4411ad2antrr 761 . . . . . . . . . 10 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐹 ⊆ (𝑁‘(𝐺𝐻)))
45 simpr1 1065 . . . . . . . . . . . 12 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝑞𝐺)
46 uncom 3740 . . . . . . . . . . . . . 14 (𝐻 ∪ {𝑞}) = ({𝑞} ∪ 𝐻)
4746uneq2i 3747 . . . . . . . . . . . . 13 ((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞})) = ((𝐺 ∖ {𝑞}) ∪ ({𝑞} ∪ 𝐻))
48 unass 3753 . . . . . . . . . . . . . 14 (((𝐺 ∖ {𝑞}) ∪ {𝑞}) ∪ 𝐻) = ((𝐺 ∖ {𝑞}) ∪ ({𝑞} ∪ 𝐻))
49 difsnid 4315 . . . . . . . . . . . . . . 15 (𝑞𝐺 → ((𝐺 ∖ {𝑞}) ∪ {𝑞}) = 𝐺)
5049uneq1d 3749 . . . . . . . . . . . . . 14 (𝑞𝐺 → (((𝐺 ∖ {𝑞}) ∪ {𝑞}) ∪ 𝐻) = (𝐺𝐻))
5148, 50syl5eqr 2669 . . . . . . . . . . . . 13 (𝑞𝐺 → ((𝐺 ∖ {𝑞}) ∪ ({𝑞} ∪ 𝐻)) = (𝐺𝐻))
5247, 51syl5eq 2667 . . . . . . . . . . . 12 (𝑞𝐺 → ((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞})) = (𝐺𝐻))
5345, 52syl 17 . . . . . . . . . . 11 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞})) = (𝐺𝐻))
5453fveq2d 6157 . . . . . . . . . 10 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝑁‘((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞}))) = (𝑁‘(𝐺𝐻)))
5544, 54sseqtr4d 3626 . . . . . . . . 9 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐹 ⊆ (𝑁‘((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞}))))
5655ssdifssd 3731 . . . . . . . 8 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ∖ {𝑟}) ⊆ (𝑁‘((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞}))))
57 simpr3 1067 . . . . . . . 8 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)
58 mreexexlem4d.B . . . . . . . . . 10 (𝜑 → (𝐹 ≈ suc 𝐿𝐺 ≈ suc 𝐿))
5958ad2antrr 761 . . . . . . . . 9 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ≈ suc 𝐿𝐺 ≈ suc 𝐿))
60 mreexexlem4d.9 . . . . . . . . . . . 12 (𝜑𝐿 ∈ ω)
6160ad2antrr 761 . . . . . . . . . . 11 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐿 ∈ ω)
62 simplr 791 . . . . . . . . . . 11 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝑟𝐹)
63 3anan12 1049 . . . . . . . . . . . . 13 ((𝐿 ∈ ω ∧ 𝐹 ≈ suc 𝐿𝑟𝐹) ↔ (𝐹 ≈ suc 𝐿 ∧ (𝐿 ∈ ω ∧ 𝑟𝐹)))
64 dif1en 8145 . . . . . . . . . . . . 13 ((𝐿 ∈ ω ∧ 𝐹 ≈ suc 𝐿𝑟𝐹) → (𝐹 ∖ {𝑟}) ≈ 𝐿)
6563, 64sylbir 225 . . . . . . . . . . . 12 ((𝐹 ≈ suc 𝐿 ∧ (𝐿 ∈ ω ∧ 𝑟𝐹)) → (𝐹 ∖ {𝑟}) ≈ 𝐿)
6665expcom 451 . . . . . . . . . . 11 ((𝐿 ∈ ω ∧ 𝑟𝐹) → (𝐹 ≈ suc 𝐿 → (𝐹 ∖ {𝑟}) ≈ 𝐿))
6761, 62, 66syl2anc 692 . . . . . . . . . 10 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ≈ suc 𝐿 → (𝐹 ∖ {𝑟}) ≈ 𝐿))
68 3anan12 1049 . . . . . . . . . . . . 13 ((𝐿 ∈ ω ∧ 𝐺 ≈ suc 𝐿𝑞𝐺) ↔ (𝐺 ≈ suc 𝐿 ∧ (𝐿 ∈ ω ∧ 𝑞𝐺)))
69 dif1en 8145 . . . . . . . . . . . . 13 ((𝐿 ∈ ω ∧ 𝐺 ≈ suc 𝐿𝑞𝐺) → (𝐺 ∖ {𝑞}) ≈ 𝐿)
7068, 69sylbir 225 . . . . . . . . . . . 12 ((𝐺 ≈ suc 𝐿 ∧ (𝐿 ∈ ω ∧ 𝑞𝐺)) → (𝐺 ∖ {𝑞}) ≈ 𝐿)
7170expcom 451 . . . . . . . . . . 11 ((𝐿 ∈ ω ∧ 𝑞𝐺) → (𝐺 ≈ suc 𝐿 → (𝐺 ∖ {𝑞}) ≈ 𝐿))
7261, 45, 71syl2anc 692 . . . . . . . . . 10 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐺 ≈ suc 𝐿 → (𝐺 ∖ {𝑞}) ≈ 𝐿))
7367, 72orim12d 882 . . . . . . . . 9 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ((𝐹 ≈ suc 𝐿𝐺 ≈ suc 𝐿) → ((𝐹 ∖ {𝑟}) ≈ 𝐿 ∨ (𝐺 ∖ {𝑞}) ≈ 𝐿)))
7459, 73mpd 15 . . . . . . . 8 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ((𝐹 ∖ {𝑟}) ≈ 𝐿 ∨ (𝐺 ∖ {𝑞}) ≈ 𝐿))
75 mreexexlem4d.A . . . . . . . . 9 (𝜑 → ∀𝑓 ∈ 𝒫 (𝑋)∀𝑔 ∈ 𝒫 (𝑋)(((𝑓𝐿𝑔𝐿) ∧ 𝑓 ⊆ (𝑁‘(𝑔)) ∧ (𝑓) ∈ 𝐼) → ∃𝑗 ∈ 𝒫 𝑔(𝑓𝑗 ∧ (𝑗) ∈ 𝐼)))
7675ad2antrr 761 . . . . . . . 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)))
7731, 40, 43, 56, 57, 74, 76mreexexlemd 16236 . . . . . . 7 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ∃𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞})((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))
78 simprl 793 . . . . . . . . . . . 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}))
7978elpwid 4146 . . . . . . . . . . 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}))
8079difss2d 3723 . . . . . . . . . 10 ((((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝑖𝐺)
81 simplr1 1101 . . . . . . . . . . 11 ((((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝑞𝐺)
8281snssd 4314 . . . . . . . . . 10 ((((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → {𝑞} ⊆ 𝐺)
8380, 82unssd 3772 . . . . . . . . 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)
8431adantr 481 . . . . . . . . . . 11 ((((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝑋 ∈ V)
859ad3antrrr 765 . . . . . . . . . . . 12 ((((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝐺 ⊆ (𝑋𝐻))
8685difss2d 3723 . . . . . . . . . . 11 ((((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝐺𝑋)
8784, 86ssexd 4770 . . . . . . . . . 10 ((((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝐺 ∈ V)
88 elpw2g 4792 . . . . . . . . . 10 (𝐺 ∈ V → ((𝑖 ∪ {𝑞}) ∈ 𝒫 𝐺 ↔ (𝑖 ∪ {𝑞}) ⊆ 𝐺))
8987, 88syl 17 . . . . . . . . 9 ((((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 ↔ (𝑖 ∪ {𝑞}) ⊆ 𝐺))
9083, 89mpbird 247 . . . . . . . 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)
91 difsnid 4315 . . . . . . . . . 10 (𝑟𝐹 → ((𝐹 ∖ {𝑟}) ∪ {𝑟}) = 𝐹)
9291ad3antlr 766 . . . . . . . . 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)
93 simprrl 803 . . . . . . . . . 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)
94 vex 3192 . . . . . . . . . . . 12 𝑟 ∈ V
95 vex 3192 . . . . . . . . . . . 12 𝑞 ∈ V
96 en2sn 7989 . . . . . . . . . . . 12 ((𝑟 ∈ V ∧ 𝑞 ∈ V) → {𝑟} ≈ {𝑞})
9794, 95, 96mp2an 707 . . . . . . . . . . 11 {𝑟} ≈ {𝑞}
9897a1i 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})
99 incom 3788 . . . . . . . . . . . 12 ((𝐹 ∖ {𝑟}) ∩ {𝑟}) = ({𝑟} ∩ (𝐹 ∖ {𝑟}))
100 disjdif 4017 . . . . . . . . . . . 12 ({𝑟} ∩ (𝐹 ∖ {𝑟})) = ∅
10199, 100eqtri 2643 . . . . . . . . . . 11 ((𝐹 ∖ {𝑟}) ∩ {𝑟}) = ∅
102101a1i 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}) = ∅)
103 ssdifin0 4027 . . . . . . . . . . 11 (𝑖 ⊆ (𝐺 ∖ {𝑞}) → (𝑖 ∩ {𝑞}) = ∅)
10479, 103syl 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}) = ∅)
105 unen 7992 . . . . . . . . . 10 ((((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ {𝑟} ≈ {𝑞}) ∧ (((𝐹 ∖ {𝑟}) ∩ {𝑟}) = ∅ ∧ (𝑖 ∩ {𝑞}) = ∅)) → ((𝐹 ∖ {𝑟}) ∪ {𝑟}) ≈ (𝑖 ∪ {𝑞}))
10693, 98, 102, 104, 105syl22anc 1324 . . . . . . . . 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}))
10792, 106eqbrtrrd 4642 . . . . . . . 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}))
108 unass 3753 . . . . . . . . . 10 ((𝑖 ∪ {𝑞}) ∪ 𝐻) = (𝑖 ∪ ({𝑞} ∪ 𝐻))
109 uncom 3740 . . . . . . . . . . 11 ({𝑞} ∪ 𝐻) = (𝐻 ∪ {𝑞})
110109uneq2i 3747 . . . . . . . . . 10 (𝑖 ∪ ({𝑞} ∪ 𝐻)) = (𝑖 ∪ (𝐻 ∪ {𝑞}))
111108, 110eqtr2i 2644 . . . . . . . . 9 (𝑖 ∪ (𝐻 ∪ {𝑞})) = ((𝑖 ∪ {𝑞}) ∪ 𝐻)
112 simprrr 804 . . . . . . . . 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)
113111, 112syl5eqelr 2703 . . . . . . . 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)
114 breq2 4622 . . . . . . . . . 10 (𝑗 = (𝑖 ∪ {𝑞}) → (𝐹𝑗𝐹 ≈ (𝑖 ∪ {𝑞})))
115 uneq1 3743 . . . . . . . . . . 11 (𝑗 = (𝑖 ∪ {𝑞}) → (𝑗𝐻) = ((𝑖 ∪ {𝑞}) ∪ 𝐻))
116115eleq1d 2683 . . . . . . . . . 10 (𝑗 = (𝑖 ∪ {𝑞}) → ((𝑗𝐻) ∈ 𝐼 ↔ ((𝑖 ∪ {𝑞}) ∪ 𝐻) ∈ 𝐼))
117114, 116anbi12d 746 . . . . . . . . 9 (𝑗 = (𝑖 ∪ {𝑞}) → ((𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼) ↔ (𝐹 ≈ (𝑖 ∪ {𝑞}) ∧ ((𝑖 ∪ {𝑞}) ∪ 𝐻) ∈ 𝐼)))
118117rspcev 3298 . . . . . . . 8 (((𝑖 ∪ {𝑞}) ∈ 𝒫 𝐺 ∧ (𝐹 ≈ (𝑖 ∪ {𝑞}) ∧ ((𝑖 ∪ {𝑞}) ∪ 𝐻) ∈ 𝐼)) → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))
11990, 107, 113, 118syl12anc 1321 . . . . . . 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))
12077, 119rexlimddv 3029 . . . . . 6 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))
12129, 120sylan2br 493 . . . . 5 (((𝜑𝑟𝐹) ∧ (𝑞𝐺 ∧ (¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))
12228, 121rexlimddv 3029 . . . 4 ((𝜑𝑟𝐹) → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))
123122adantlr 750 . . 3 (((𝜑𝐹 ≠ ∅) ∧ 𝑟𝐹) → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))
12420, 123exlimddv 1860 . 2 ((𝜑𝐹 ≠ ∅) → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))
12517, 124pm2.61dane 2877 1 (𝜑 → ∃𝑗 ∈ 𝒫 𝐺(𝐹𝑗 ∧ (𝑗𝐻) ∈ 𝐼))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1036  wal 1478   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2907  wrex 2908  Vcvv 3189  cdif 3556  cun 3557  cin 3558  wss 3559  c0 3896  𝒫 cpw 4135  {csn 4153   class class class wbr 4618  suc csuc 5689  cfv 5852  ωcom 7019  cen 7904  Moorecmre 16174  mrClscmrc 16175  mrIndcmri 16176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-om 7020  df-1o 7512  df-er 7694  df-en 7908  df-fin 7911  df-mre 16178  df-mrc 16179  df-mri 16180
This theorem is referenced by:  mreexexd  16240  mreexexdOLD  16241
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