Theorem mreexexlem4d 17705

Description: Induction step of the induction in mreexexd 17706. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
mreexexlem2d.1	⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))
mreexexlem2d.2	⊢ 𝑁 = (mrCls‘𝐴)
mreexexlem2d.3	⊢ 𝐼 = (mrInd‘𝐴)
mreexexlem2d.4	⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))
mreexexlem2d.5	⊢ (𝜑 → 𝐹 ⊆ (𝑋 ∖ 𝐻))
mreexexlem2d.6	⊢ (𝜑 → 𝐺 ⊆ (𝑋 ∖ 𝐻))
mreexexlem2d.7	⊢ (𝜑 → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻)))
mreexexlem2d.8	⊢ (𝜑 → (𝐹 ∪ 𝐻) ∈ 𝐼)
mreexexlem4d.9	⊢ (𝜑 → 𝐿 ∈ ω)
mreexexlem4d.A	⊢ (𝜑 → ∀ℎ∀𝑓 ∈ 𝒫 (𝑋 ∖ ℎ)∀𝑔 ∈ 𝒫 (𝑋 ∖ ℎ)(((𝑓 ≈ 𝐿 ∨ 𝑔 ≈ 𝐿) ∧ 𝑓 ⊆ (𝑁‘(𝑔 ∪ ℎ)) ∧ (𝑓 ∪ ℎ) ∈ 𝐼) → ∃𝑗 ∈ 𝒫 𝑔(𝑓 ≈ 𝑗 ∧ (𝑗 ∪ ℎ) ∈ 𝐼)))
mreexexlem4d.B	⊢ (𝜑 → (𝐹 ≈ suc 𝐿 ∨ 𝐺 ≈ suc 𝐿))

Assertion

Ref	Expression
mreexexlem4d	⊢ (𝜑 → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼))

Distinct variable groups:   𝑓,𝑔,ℎ,𝑋   𝑓,𝐼,𝑗,𝑔,ℎ   𝑓,𝐿,𝑔,ℎ   𝑓,𝑁,𝑔,ℎ   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,ℎ,𝑗)   𝐴(𝑦,𝑧,𝑓,𝑔,ℎ,𝑗,𝑠)   𝐹(𝑓,𝑔,ℎ)   𝐺(𝑓,𝑔,ℎ)   𝐻(𝑓,𝑔,ℎ)   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.

Step	Hyp	Ref	Expression
1		mreexexlem2d.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))
2	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐹 = ∅) → 𝐴 ∈ (Moore‘𝑋))
3		mreexexlem2d.2	. . 3 ⊢ 𝑁 = (mrCls‘𝐴)
4		mreexexlem2d.3	. . 3 ⊢ 𝐼 = (mrInd‘𝐴)
5		mreexexlem2d.4	. . . 4 ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))
6	5	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐹 = ∅) → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))
7		mreexexlem2d.5	. . . 4 ⊢ (𝜑 → 𝐹 ⊆ (𝑋 ∖ 𝐻))
8	7	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐹 = ∅) → 𝐹 ⊆ (𝑋 ∖ 𝐻))
9		mreexexlem2d.6	. . . 4 ⊢ (𝜑 → 𝐺 ⊆ (𝑋 ∖ 𝐻))
10	9	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐹 = ∅) → 𝐺 ⊆ (𝑋 ∖ 𝐻))
11		mreexexlem2d.7	. . . 4 ⊢ (𝜑 → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻)))
12	11	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐹 = ∅) → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻)))
13		mreexexlem2d.8	. . . 4 ⊢ (𝜑 → (𝐹 ∪ 𝐻) ∈ 𝐼)
14	13	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐹 = ∅) → (𝐹 ∪ 𝐻) ∈ 𝐼)
15		animorrl 981	. . 3 ⊢ ((𝜑 ∧ 𝐹 = ∅) → (𝐹 = ∅ ∨ 𝐺 = ∅))
16	2, 3, 4, 6, 8, 10, 12, 14, 15	mreexexlem3d 17704	. 2 ⊢ ((𝜑 ∧ 𝐹 = ∅) → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼))
17		n0 4376	. . . . 5 ⊢ (𝐹 ≠ ∅ ↔ ∃𝑟 𝑟 ∈ 𝐹)
18	17	biimpi 216	. . . 4 ⊢ (𝐹 ≠ ∅ → ∃𝑟 𝑟 ∈ 𝐹)
19	18	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → ∃𝑟 𝑟 ∈ 𝐹)
20	1	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝐹) → 𝐴 ∈ (Moore‘𝑋))
21	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝐹) → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))
22	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝐹) → 𝐹 ⊆ (𝑋 ∖ 𝐻))
23	9	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝐹) → 𝐺 ⊆ (𝑋 ∖ 𝐻))
24	11	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝐹) → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻)))
25	13	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝐹) → (𝐹 ∪ 𝐻) ∈ 𝐼)
26		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝐹) → 𝑟 ∈ 𝐹)
27	20, 3, 4, 21, 22, 23, 24, 25, 26	mreexexlem2d 17703	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝐹) → ∃𝑞 ∈ 𝐺 (¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))
28		3anass 1095	. . . . . 6 ⊢ ((𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼) ↔ (𝑞 ∈ 𝐺 ∧ (¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)))
29	1	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐴 ∈ (Moore‘𝑋))
30	29	elfvexd 6959	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝑋 ∈ V)
31		simpr2 1195	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}))
32		difsnb 4831	. . . . . . . . . . 11 ⊢ (¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ↔ ((𝐹 ∖ {𝑟}) ∖ {𝑞}) = (𝐹 ∖ {𝑟}))
33	31, 32	sylib 218	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ((𝐹 ∖ {𝑟}) ∖ {𝑞}) = (𝐹 ∖ {𝑟}))
34	7	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐹 ⊆ (𝑋 ∖ 𝐻))
35	34	ssdifssd 4170	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ∖ {𝑟}) ⊆ (𝑋 ∖ 𝐻))
36	35	ssdifd 4168	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ((𝐹 ∖ {𝑟}) ∖ {𝑞}) ⊆ ((𝑋 ∖ 𝐻) ∖ {𝑞}))
37	33, 36	eqsstrrd 4048	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ∖ {𝑟}) ⊆ ((𝑋 ∖ 𝐻) ∖ {𝑞}))
38		difun1 4318	. . . . . . . . 9 ⊢ (𝑋 ∖ (𝐻 ∪ {𝑞})) = ((𝑋 ∖ 𝐻) ∖ {𝑞})
39	37, 38	sseqtrrdi 4060	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ∖ {𝑟}) ⊆ (𝑋 ∖ (𝐻 ∪ {𝑞})))
40	9	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐺 ⊆ (𝑋 ∖ 𝐻))
41	40	ssdifd 4168	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐺 ∖ {𝑞}) ⊆ ((𝑋 ∖ 𝐻) ∖ {𝑞}))
42	41, 38	sseqtrrdi 4060	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐺 ∖ {𝑞}) ⊆ (𝑋 ∖ (𝐻 ∪ {𝑞})))
43	11	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻)))
44		simpr1 1194	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝑞 ∈ 𝐺)
45		uncom 4181	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∪ {𝑞}) = ({𝑞} ∪ 𝐻)
46	45	uneq2i 4188	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞})) = ((𝐺 ∖ {𝑞}) ∪ ({𝑞} ∪ 𝐻))
47		unass 4195	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∖ {𝑞}) ∪ {𝑞}) ∪ 𝐻) = ((𝐺 ∖ {𝑞}) ∪ ({𝑞} ∪ 𝐻))
48		difsnid 4835	. . . . . . . . . . . . . . 15 ⊢ (𝑞 ∈ 𝐺 → ((𝐺 ∖ {𝑞}) ∪ {𝑞}) = 𝐺)
49	48	uneq1d 4190	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ 𝐺 → (((𝐺 ∖ {𝑞}) ∪ {𝑞}) ∪ 𝐻) = (𝐺 ∪ 𝐻))
50	47, 49	eqtr3id 2794	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ 𝐺 → ((𝐺 ∖ {𝑞}) ∪ ({𝑞} ∪ 𝐻)) = (𝐺 ∪ 𝐻))
51	46, 50	eqtrid 2792	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ 𝐺 → ((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞})) = (𝐺 ∪ 𝐻))
52	44, 51	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞})) = (𝐺 ∪ 𝐻))
53	52	fveq2d 6924	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝑁‘((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞}))) = (𝑁‘(𝐺 ∪ 𝐻)))
54	43, 53	sseqtrrd 4050	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐹 ⊆ (𝑁‘((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞}))))
55	54	ssdifssd 4170	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ∖ {𝑟}) ⊆ (𝑁‘((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞}))))
56		simpr3 1196	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)
57		mreexexlem4d.B	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ≈ suc 𝐿 ∨ 𝐺 ≈ suc 𝐿))
58	57	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ≈ suc 𝐿 ∨ 𝐺 ≈ suc 𝐿))
59		mreexexlem4d.9	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐿 ∈ ω)
60	59	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐿 ∈ ω)
61		simplr 768	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝑟 ∈ 𝐹)
62		3anan12 1096	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ ω ∧ 𝐹 ≈ suc 𝐿 ∧ 𝑟 ∈ 𝐹) ↔ (𝐹 ≈ suc 𝐿 ∧ (𝐿 ∈ ω ∧ 𝑟 ∈ 𝐹)))
63		dif1ennn 9227	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ ω ∧ 𝐹 ≈ suc 𝐿 ∧ 𝑟 ∈ 𝐹) → (𝐹 ∖ {𝑟}) ≈ 𝐿)
64	62, 63	sylbir 235	. . . . . . . . . . . 12 ⊢ ((𝐹 ≈ suc 𝐿 ∧ (𝐿 ∈ ω ∧ 𝑟 ∈ 𝐹)) → (𝐹 ∖ {𝑟}) ≈ 𝐿)
65	64	expcom 413	. . . . . . . . . . 11 ⊢ ((𝐿 ∈ ω ∧ 𝑟 ∈ 𝐹) → (𝐹 ≈ suc 𝐿 → (𝐹 ∖ {𝑟}) ≈ 𝐿))
66	60, 61, 65	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ≈ suc 𝐿 → (𝐹 ∖ {𝑟}) ≈ 𝐿))
67		3anan12 1096	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ ω ∧ 𝐺 ≈ suc 𝐿 ∧ 𝑞 ∈ 𝐺) ↔ (𝐺 ≈ suc 𝐿 ∧ (𝐿 ∈ ω ∧ 𝑞 ∈ 𝐺)))
68		dif1ennn 9227	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ ω ∧ 𝐺 ≈ suc 𝐿 ∧ 𝑞 ∈ 𝐺) → (𝐺 ∖ {𝑞}) ≈ 𝐿)
69	67, 68	sylbir 235	. . . . . . . . . . . 12 ⊢ ((𝐺 ≈ suc 𝐿 ∧ (𝐿 ∈ ω ∧ 𝑞 ∈ 𝐺)) → (𝐺 ∖ {𝑞}) ≈ 𝐿)
70	69	expcom 413	. . . . . . . . . . 11 ⊢ ((𝐿 ∈ ω ∧ 𝑞 ∈ 𝐺) → (𝐺 ≈ suc 𝐿 → (𝐺 ∖ {𝑞}) ≈ 𝐿))
71	60, 44, 70	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐺 ≈ suc 𝐿 → (𝐺 ∖ {𝑞}) ≈ 𝐿))
72	66, 71	orim12d 965	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ((𝐹 ≈ suc 𝐿 ∨ 𝐺 ≈ suc 𝐿) → ((𝐹 ∖ {𝑟}) ≈ 𝐿 ∨ (𝐺 ∖ {𝑞}) ≈ 𝐿)))
73	58, 72	mpd 15	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ((𝐹 ∖ {𝑟}) ≈ 𝐿 ∨ (𝐺 ∖ {𝑞}) ≈ 𝐿))
74		mreexexlem4d.A	. . . . . . . . 9 ⊢ (𝜑 → ∀ℎ∀𝑓 ∈ 𝒫 (𝑋 ∖ ℎ)∀𝑔 ∈ 𝒫 (𝑋 ∖ ℎ)(((𝑓 ≈ 𝐿 ∨ 𝑔 ≈ 𝐿) ∧ 𝑓 ⊆ (𝑁‘(𝑔 ∪ ℎ)) ∧ (𝑓 ∪ ℎ) ∈ 𝐼) → ∃𝑗 ∈ 𝒫 𝑔(𝑓 ≈ 𝑗 ∧ (𝑗 ∪ ℎ) ∈ 𝐼)))
75	74	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ∀ℎ∀𝑓 ∈ 𝒫 (𝑋 ∖ ℎ)∀𝑔 ∈ 𝒫 (𝑋 ∖ ℎ)(((𝑓 ≈ 𝐿 ∨ 𝑔 ≈ 𝐿) ∧ 𝑓 ⊆ (𝑁‘(𝑔 ∪ ℎ)) ∧ (𝑓 ∪ ℎ) ∈ 𝐼) → ∃𝑗 ∈ 𝒫 𝑔(𝑓 ≈ 𝑗 ∧ (𝑗 ∪ ℎ) ∈ 𝐼)))
76	30, 39, 42, 55, 56, 73, 75	mreexexlemd 17702	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ∃𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞})((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))
77	30	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝑋 ∈ V)
78	9	ad3antrrr 729	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝐺 ⊆ (𝑋 ∖ 𝐻))
79	78	difss2d 4162	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝐺 ⊆ 𝑋)
80	77, 79	ssexd 5342	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝐺 ∈ V)
81		simprl 770	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}))
82	81	elpwid 4631	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝑖 ⊆ (𝐺 ∖ {𝑞}))
83	82	difss2d 4162	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝑖 ⊆ 𝐺)
84		simplr1 1215	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝑞 ∈ 𝐺)
85	84	snssd 4834	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → {𝑞} ⊆ 𝐺)
86	83, 85	unssd 4215	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → (𝑖 ∪ {𝑞}) ⊆ 𝐺)
87	80, 86	sselpwd 5346	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 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 4835	. . . . . . . . . 10 ⊢ (𝑟 ∈ 𝐹 → ((𝐹 ∖ {𝑟}) ∪ {𝑟}) = 𝐹)
89	88	ad3antlr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 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 780	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → (𝐹 ∖ {𝑟}) ≈ 𝑖)
91		en2sn 9106	. . . . . . . . . . . 12 ⊢ ((𝑟 ∈ V ∧ 𝑞 ∈ V) → {𝑟} ≈ {𝑞})
92	91	el2v 3495	. . . . . . . . . . 11 ⊢ {𝑟} ≈ {𝑞}
93	92	a1i 11	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → {𝑟} ≈ {𝑞})
94		disjdifr 4496	. . . . . . . . . . 11 ⊢ ((𝐹 ∖ {𝑟}) ∩ {𝑟}) = ∅
95	94	a1i 11	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → ((𝐹 ∖ {𝑟}) ∩ {𝑟}) = ∅)
96		ssdifin0 4509	. . . . . . . . . . 11 ⊢ (𝑖 ⊆ (𝐺 ∖ {𝑞}) → (𝑖 ∩ {𝑞}) = ∅)
97	82, 96	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → (𝑖 ∩ {𝑞}) = ∅)
98		unen 9112	. . . . . . . . . 10 ⊢ ((((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ {𝑟} ≈ {𝑞}) ∧ (((𝐹 ∖ {𝑟}) ∩ {𝑟}) = ∅ ∧ (𝑖 ∩ {𝑞}) = ∅)) → ((𝐹 ∖ {𝑟}) ∪ {𝑟}) ≈ (𝑖 ∪ {𝑞}))
99	90, 93, 95, 97, 98	syl22anc 838	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → ((𝐹 ∖ {𝑟}) ∪ {𝑟}) ≈ (𝑖 ∪ {𝑞}))
100	89, 99	eqbrtrrd 5190	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝐹 ≈ (𝑖 ∪ {𝑞}))
101		unass 4195	. . . . . . . . . 10 ⊢ ((𝑖 ∪ {𝑞}) ∪ 𝐻) = (𝑖 ∪ ({𝑞} ∪ 𝐻))
102		uncom 4181	. . . . . . . . . . 11 ⊢ ({𝑞} ∪ 𝐻) = (𝐻 ∪ {𝑞})
103	102	uneq2i 4188	. . . . . . . . . 10 ⊢ (𝑖 ∪ ({𝑞} ∪ 𝐻)) = (𝑖 ∪ (𝐻 ∪ {𝑞}))
104	101, 103	eqtr2i 2769	. . . . . . . . 9 ⊢ (𝑖 ∪ (𝐻 ∪ {𝑞})) = ((𝑖 ∪ {𝑞}) ∪ 𝐻)
105		simprrr 781	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)
106	104, 105	eqeltrrid 2849	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 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 5170	. . . . . . . . . 10 ⊢ (𝑗 = (𝑖 ∪ {𝑞}) → (𝐹 ≈ 𝑗 ↔ 𝐹 ≈ (𝑖 ∪ {𝑞})))
108		uneq1 4184	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑖 ∪ {𝑞}) → (𝑗 ∪ 𝐻) = ((𝑖 ∪ {𝑞}) ∪ 𝐻))
109	108	eleq1d 2829	. . . . . . . . . 10 ⊢ (𝑗 = (𝑖 ∪ {𝑞}) → ((𝑗 ∪ 𝐻) ∈ 𝐼 ↔ ((𝑖 ∪ {𝑞}) ∪ 𝐻) ∈ 𝐼))
110	107, 109	anbi12d 631	. . . . . . . . 9 ⊢ (𝑗 = (𝑖 ∪ {𝑞}) → ((𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼) ↔ (𝐹 ≈ (𝑖 ∪ {𝑞}) ∧ ((𝑖 ∪ {𝑞}) ∪ 𝐻) ∈ 𝐼)))
111	110	rspcev 3635	. . . . . . . 8 ⊢ (((𝑖 ∪ {𝑞}) ∈ 𝒫 𝐺 ∧ (𝐹 ≈ (𝑖 ∪ {𝑞}) ∧ ((𝑖 ∪ {𝑞}) ∪ 𝐻) ∈ 𝐼)) → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼))
112	87, 100, 106, 111	syl12anc 836	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼))
113	76, 112	rexlimddv 3167	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼))
114	28, 113	sylan2br 594	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ (¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼))
115	27, 114	rexlimddv 3167	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝐹) → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼))
116	115	adantlr 714	. . 3 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ 𝑟 ∈ 𝐹) → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼))
117	19, 116	exlimddv 1934	. 2 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼))
118	16, 117	pm2.61dane 3035	1 ⊢ (𝜑 → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 846   ∧ w3a 1087  ∀wal 1535   = wceq 1537  ∃wex 1777   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076  Vcvv 3488   ∖ cdif 3973   ∪ cun 3974   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  𝒫 cpw 4622  {csn 4648   class class class wbr 5166  suc csuc 6397  ‘cfv 6573  ωcom 7903   ≈ cen 9000  Moorecmre 17640  mrClscmrc 17641  mrIndcmri 17642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-om 7904  df-en 9004  df-mre 17644  df-mrc 17645  df-mri 17646

This theorem is referenced by:  mreexexd  17706