Theorem mreexexlem4d 16910

Description: Induction step of the induction in mreexexd 16911. (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 484	. . 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 484	. . 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 484	. . 3 ⊢ ((𝜑 ∧ 𝐹 = ∅) → 𝐹 ⊆ (𝑋 ∖ 𝐻))
9		mreexexlem2d.6	. . . 4 ⊢ (𝜑 → 𝐺 ⊆ (𝑋 ∖ 𝐻))
10	9	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐹 = ∅) → 𝐺 ⊆ (𝑋 ∖ 𝐻))
11		mreexexlem2d.7	. . . 4 ⊢ (𝜑 → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻)))
12	11	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐹 = ∅) → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻)))
13		mreexexlem2d.8	. . . 4 ⊢ (𝜑 → (𝐹 ∪ 𝐻) ∈ 𝐼)
14	13	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐹 = ∅) → (𝐹 ∪ 𝐻) ∈ 𝐼)
15		animorrl 978	. . 3 ⊢ ((𝜑 ∧ 𝐹 = ∅) → (𝐹 = ∅ ∨ 𝐺 = ∅))
16	2, 3, 4, 6, 8, 10, 12, 14, 15	mreexexlem3d 16909	. 2 ⊢ ((𝜑 ∧ 𝐹 = ∅) → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼))
17		n0 4260	. . . . 5 ⊢ (𝐹 ≠ ∅ ↔ ∃𝑟 𝑟 ∈ 𝐹)
18	17	biimpi 219	. . . 4 ⊢ (𝐹 ≠ ∅ → ∃𝑟 𝑟 ∈ 𝐹)
19	18	adantl 485	. . 3 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → ∃𝑟 𝑟 ∈ 𝐹)
20	1	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝐹) → 𝐴 ∈ (Moore‘𝑋))
21	5	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝐹) → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))
22	7	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝐹) → 𝐹 ⊆ (𝑋 ∖ 𝐻))
23	9	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝐹) → 𝐺 ⊆ (𝑋 ∖ 𝐻))
24	11	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝐹) → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻)))
25	13	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝐹) → (𝐹 ∪ 𝐻) ∈ 𝐼)
26		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝐹) → 𝑟 ∈ 𝐹)
27	20, 3, 4, 21, 22, 23, 24, 25, 26	mreexexlem2d 16908	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝐹) → ∃𝑞 ∈ 𝐺 (¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))
28		3anass 1092	. . . . . 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 6679	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝑋 ∈ V)
31		simpr2 1192	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}))
32		difsnb 4699	. . . . . . . . . . 11 ⊢ (¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ↔ ((𝐹 ∖ {𝑟}) ∖ {𝑞}) = (𝐹 ∖ {𝑟}))
33	31, 32	sylib 221	. . . . . . . . . 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 4070	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ∖ {𝑟}) ⊆ (𝑋 ∖ 𝐻))
36	35	ssdifd 4068	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ((𝐹 ∖ {𝑟}) ∖ {𝑞}) ⊆ ((𝑋 ∖ 𝐻) ∖ {𝑞}))
37	33, 36	eqsstrrd 3954	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ∖ {𝑟}) ⊆ ((𝑋 ∖ 𝐻) ∖ {𝑞}))
38		difun1 4214	. . . . . . . . 9 ⊢ (𝑋 ∖ (𝐻 ∪ {𝑞})) = ((𝑋 ∖ 𝐻) ∖ {𝑞})
39	37, 38	sseqtrrdi 3966	. . . . . . . 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 4068	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐺 ∖ {𝑞}) ⊆ ((𝑋 ∖ 𝐻) ∖ {𝑞}))
42	41, 38	sseqtrrdi 3966	. . . . . . . 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 1191	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝑞 ∈ 𝐺)
45		uncom 4080	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∪ {𝑞}) = ({𝑞} ∪ 𝐻)
46	45	uneq2i 4087	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞})) = ((𝐺 ∖ {𝑞}) ∪ ({𝑞} ∪ 𝐻))
47		unass 4093	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∖ {𝑞}) ∪ {𝑞}) ∪ 𝐻) = ((𝐺 ∖ {𝑞}) ∪ ({𝑞} ∪ 𝐻))
48		difsnid 4703	. . . . . . . . . . . . . . 15 ⊢ (𝑞 ∈ 𝐺 → ((𝐺 ∖ {𝑞}) ∪ {𝑞}) = 𝐺)
49	48	uneq1d 4089	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ 𝐺 → (((𝐺 ∖ {𝑞}) ∪ {𝑞}) ∪ 𝐻) = (𝐺 ∪ 𝐻))
50	47, 49	syl5eqr 2847	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ 𝐺 → ((𝐺 ∖ {𝑞}) ∪ ({𝑞} ∪ 𝐻)) = (𝐺 ∪ 𝐻))
51	46, 50	syl5eq 2845	. . . . . . . . . . . 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 6649	. . . . . . . . . 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 3956	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐹 ⊆ (𝑁‘((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞}))))
55	54	ssdifssd 4070	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ∖ {𝑟}) ⊆ (𝑁‘((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞}))))
56		simpr3 1193	. . . . . . . 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 1093	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ ω ∧ 𝐹 ≈ suc 𝐿 ∧ 𝑟 ∈ 𝐹) ↔ (𝐹 ≈ suc 𝐿 ∧ (𝐿 ∈ ω ∧ 𝑟 ∈ 𝐹)))
63		dif1en 8735	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ ω ∧ 𝐹 ≈ suc 𝐿 ∧ 𝑟 ∈ 𝐹) → (𝐹 ∖ {𝑟}) ≈ 𝐿)
64	62, 63	sylbir 238	. . . . . . . . . . . 12 ⊢ ((𝐹 ≈ suc 𝐿 ∧ (𝐿 ∈ ω ∧ 𝑟 ∈ 𝐹)) → (𝐹 ∖ {𝑟}) ≈ 𝐿)
65	64	expcom 417	. . . . . . . . . . 11 ⊢ ((𝐿 ∈ ω ∧ 𝑟 ∈ 𝐹) → (𝐹 ≈ suc 𝐿 → (𝐹 ∖ {𝑟}) ≈ 𝐿))
66	60, 61, 65	syl2anc 587	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ≈ suc 𝐿 → (𝐹 ∖ {𝑟}) ≈ 𝐿))
67		3anan12 1093	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ ω ∧ 𝐺 ≈ suc 𝐿 ∧ 𝑞 ∈ 𝐺) ↔ (𝐺 ≈ suc 𝐿 ∧ (𝐿 ∈ ω ∧ 𝑞 ∈ 𝐺)))
68		dif1en 8735	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ ω ∧ 𝐺 ≈ suc 𝐿 ∧ 𝑞 ∈ 𝐺) → (𝐺 ∖ {𝑞}) ≈ 𝐿)
69	67, 68	sylbir 238	. . . . . . . . . . . 12 ⊢ ((𝐺 ≈ suc 𝐿 ∧ (𝐿 ∈ ω ∧ 𝑞 ∈ 𝐺)) → (𝐺 ∖ {𝑞}) ≈ 𝐿)
70	69	expcom 417	. . . . . . . . . . 11 ⊢ ((𝐿 ∈ ω ∧ 𝑞 ∈ 𝐺) → (𝐺 ≈ suc 𝐿 → (𝐺 ∖ {𝑞}) ≈ 𝐿))
71	60, 44, 70	syl2anc 587	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐺 ≈ suc 𝐿 → (𝐺 ∖ {𝑞}) ≈ 𝐿))
72	66, 71	orim12d 962	. . . . . . . . 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 16907	. . . . . . 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 484	. . . . . . . . . 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 4062	. . . . . . . . . 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 5192	. . . . . . . . 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 4508	. . . . . . . . . . 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 4062	. . . . . . . . . 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 1212	. . . . . . . . . . 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 4702	. . . . . . . . . 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 4113	. . . . . . . . 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 5194	. . . . . . . 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 4703	. . . . . . . . . 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 8576	. . . . . . . . . . . 12 ⊢ ((𝑟 ∈ V ∧ 𝑞 ∈ V) → {𝑟} ≈ {𝑞})
92	91	el2v 3448	. . . . . . . . . . 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		incom 4128	. . . . . . . . . . . 12 ⊢ ((𝐹 ∖ {𝑟}) ∩ {𝑟}) = ({𝑟} ∩ (𝐹 ∖ {𝑟}))
95		disjdif 4379	. . . . . . . . . . . 12 ⊢ ({𝑟} ∩ (𝐹 ∖ {𝑟})) = ∅
96	94, 95	eqtri 2821	. . . . . . . . . . 11 ⊢ ((𝐹 ∖ {𝑟}) ∩ {𝑟}) = ∅
97	96	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}) = ∅)
98		ssdifin0 4389	. . . . . . . . . . 11 ⊢ (𝑖 ⊆ (𝐺 ∖ {𝑞}) → (𝑖 ∩ {𝑞}) = ∅)
99	82, 98	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}) = ∅)
100		unen 8579	. . . . . . . . . 10 ⊢ ((((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ {𝑟} ≈ {𝑞}) ∧ (((𝐹 ∖ {𝑟}) ∩ {𝑟}) = ∅ ∧ (𝑖 ∩ {𝑞}) = ∅)) → ((𝐹 ∖ {𝑟}) ∪ {𝑟}) ≈ (𝑖 ∪ {𝑞}))
101	90, 93, 97, 99, 100	syl22anc 837	. . . . . . . . 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}))
102	89, 101	eqbrtrrd 5054	. . . . . . . 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}))
103		unass 4093	. . . . . . . . . 10 ⊢ ((𝑖 ∪ {𝑞}) ∪ 𝐻) = (𝑖 ∪ ({𝑞} ∪ 𝐻))
104		uncom 4080	. . . . . . . . . . 11 ⊢ ({𝑞} ∪ 𝐻) = (𝐻 ∪ {𝑞})
105	104	uneq2i 4087	. . . . . . . . . 10 ⊢ (𝑖 ∪ ({𝑞} ∪ 𝐻)) = (𝑖 ∪ (𝐻 ∪ {𝑞}))
106	103, 105	eqtr2i 2822	. . . . . . . . 9 ⊢ (𝑖 ∪ (𝐻 ∪ {𝑞})) = ((𝑖 ∪ {𝑞}) ∪ 𝐻)
107		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)
108	106, 107	eqeltrrid 2895	. . . . . . . 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)
109		breq2 5034	. . . . . . . . . 10 ⊢ (𝑗 = (𝑖 ∪ {𝑞}) → (𝐹 ≈ 𝑗 ↔ 𝐹 ≈ (𝑖 ∪ {𝑞})))
110		uneq1 4083	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑖 ∪ {𝑞}) → (𝑗 ∪ 𝐻) = ((𝑖 ∪ {𝑞}) ∪ 𝐻))
111	110	eleq1d 2874	. . . . . . . . . 10 ⊢ (𝑗 = (𝑖 ∪ {𝑞}) → ((𝑗 ∪ 𝐻) ∈ 𝐼 ↔ ((𝑖 ∪ {𝑞}) ∪ 𝐻) ∈ 𝐼))
112	109, 111	anbi12d 633	. . . . . . . . 9 ⊢ (𝑗 = (𝑖 ∪ {𝑞}) → ((𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼) ↔ (𝐹 ≈ (𝑖 ∪ {𝑞}) ∧ ((𝑖 ∪ {𝑞}) ∪ 𝐻) ∈ 𝐼)))
113	112	rspcev 3571	. . . . . . . 8 ⊢ (((𝑖 ∪ {𝑞}) ∈ 𝒫 𝐺 ∧ (𝐹 ≈ (𝑖 ∪ {𝑞}) ∧ ((𝑖 ∪ {𝑞}) ∪ 𝐻) ∈ 𝐼)) → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼))
114	87, 102, 108, 113	syl12anc 835	. . . . . . 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))
115	76, 114	rexlimddv 3250	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼))
116	28, 115	sylan2br 597	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ (¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼))
117	27, 116	rexlimddv 3250	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝐹) → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼))
118	117	adantlr 714	. . 3 ⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ 𝑟 ∈ 𝐹) → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼))
119	19, 118	exlimddv 1936	. 2 ⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼))
120	16, 119	pm2.61dane 3074	1 ⊢ (𝜑 → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 844   ∧ w3a 1084  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ∖ cdif 3878   ∪ cun 3879   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  𝒫 cpw 4497  {csn 4525   class class class wbr 5030  suc csuc 6161  ‘cfv 6324  ωcom 7560   ≈ cen 8489  Moorecmre 16845  mrClscmrc 16846  mrIndcmri 16847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-om 7561  df-1o 8085  df-er 8272  df-en 8493  df-fin 8496  df-mre 16849  df-mrc 16850  df-mri 16851

This theorem is referenced by:  mreexexd  16911