Theorem mreexexlem3d 17355

Description: Base case of the induction in mreexexd 17357. (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)
mreexexlem3d.9	⊢ (𝜑 → (𝐹 = ∅ ∨ 𝐺 = ∅))

Assertion

Ref	Expression
mreexexlem3d	⊢ (𝜑 → ∃𝑖 ∈ 𝒫 𝐺(𝐹 ≈ 𝑖 ∧ (𝑖 ∪ 𝐻) ∈ 𝐼))

Distinct variable groups:   𝑖,𝐹   𝑖,𝐺   𝑖,𝐻   𝑖,𝐼

Allowed substitution hints:   𝜑(𝑦,𝑧,𝑖,𝑠)   𝐴(𝑦,𝑧,𝑖,𝑠)   𝐹(𝑦,𝑧,𝑠)   𝐺(𝑦,𝑧,𝑠)   𝐻(𝑦,𝑧,𝑠)   𝐼(𝑦,𝑧,𝑠)   𝑁(𝑦,𝑧,𝑖,𝑠)   𝑋(𝑦,𝑧,𝑖,𝑠)

Proof of Theorem mreexexlem3d

Step	Hyp	Ref	Expression
1		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = ∅) → 𝐹 = ∅)
2		mreexexlem2d.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))
3	2	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐴 ∈ (Moore‘𝑋))
4		mreexexlem2d.2	. . . . . . . . 9 ⊢ 𝑁 = (mrCls‘𝐴)
5		mreexexlem2d.3	. . . . . . . . 9 ⊢ 𝐼 = (mrInd‘𝐴)
6		mreexexlem2d.7	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻)))
7	6	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻)))
8		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐺 = ∅)
9	8	uneq1d 4096	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐺 = ∅) → (𝐺 ∪ 𝐻) = (∅ ∪ 𝐻))
10		uncom 4087	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∪ ∅) = (∅ ∪ 𝐻)
11		un0 4324	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∪ ∅) = 𝐻
12	10, 11	eqtr3i 2768	. . . . . . . . . . . . 13 ⊢ (∅ ∪ 𝐻) = 𝐻
13	9, 12	eqtrdi 2794	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐺 = ∅) → (𝐺 ∪ 𝐻) = 𝐻)
14	13	fveq2d 6778	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐺 = ∅) → (𝑁‘(𝐺 ∪ 𝐻)) = (𝑁‘𝐻))
15	7, 14	sseqtrd 3961	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐹 ⊆ (𝑁‘𝐻))
16		mreexexlem2d.8	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 ∪ 𝐻) ∈ 𝐼)
17	16	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐺 = ∅) → (𝐹 ∪ 𝐻) ∈ 𝐼)
18	5, 3, 17	mrissd 17345	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐺 = ∅) → (𝐹 ∪ 𝐻) ⊆ 𝑋)
19	18	unssbd 4122	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐻 ⊆ 𝑋)
20	3, 4, 19	mrcssidd 17334	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐻 ⊆ (𝑁‘𝐻))
21	15, 20	unssd 4120	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐺 = ∅) → (𝐹 ∪ 𝐻) ⊆ (𝑁‘𝐻))
22		ssun2 4107	. . . . . . . . . 10 ⊢ 𝐻 ⊆ (𝐹 ∪ 𝐻)
23	22	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐻 ⊆ (𝐹 ∪ 𝐻))
24	3, 4, 5, 21, 23, 17	mrissmrcd 17349	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐺 = ∅) → (𝐹 ∪ 𝐻) = 𝐻)
25		ssequn1 4114	. . . . . . . 8 ⊢ (𝐹 ⊆ 𝐻 ↔ (𝐹 ∪ 𝐻) = 𝐻)
26	24, 25	sylibr 233	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐹 ⊆ 𝐻)
27		mreexexlem2d.5	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ⊆ (𝑋 ∖ 𝐻))
28	27	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐹 ⊆ (𝑋 ∖ 𝐻))
29	26, 28	ssind 4166	. . . . . 6 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐹 ⊆ (𝐻 ∩ (𝑋 ∖ 𝐻)))
30		disjdif 4405	. . . . . 6 ⊢ (𝐻 ∩ (𝑋 ∖ 𝐻)) = ∅
31	29, 30	sseqtrdi 3971	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐹 ⊆ ∅)
32		ss0b 4331	. . . . 5 ⊢ (𝐹 ⊆ ∅ ↔ 𝐹 = ∅)
33	31, 32	sylib 217	. . . 4 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐹 = ∅)
34		mreexexlem3d.9	. . . 4 ⊢ (𝜑 → (𝐹 = ∅ ∨ 𝐺 = ∅))
35	1, 33, 34	mpjaodan 956	. . 3 ⊢ (𝜑 → 𝐹 = ∅)
36		0elpw 5278	. . 3 ⊢ ∅ ∈ 𝒫 𝐺
37	35, 36	eqeltrdi 2847	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝒫 𝐺)
38	2	elfvexd 6808	. . . 4 ⊢ (𝜑 → 𝑋 ∈ V)
39	27	difss2d 4069	. . . 4 ⊢ (𝜑 → 𝐹 ⊆ 𝑋)
40	38, 39	ssexd 5248	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
41		enrefg 8772	. . 3 ⊢ (𝐹 ∈ V → 𝐹 ≈ 𝐹)
42	40, 41	syl 17	. 2 ⊢ (𝜑 → 𝐹 ≈ 𝐹)
43		breq2 5078	. . . 4 ⊢ (𝑖 = 𝐹 → (𝐹 ≈ 𝑖 ↔ 𝐹 ≈ 𝐹))
44		uneq1 4090	. . . . 5 ⊢ (𝑖 = 𝐹 → (𝑖 ∪ 𝐻) = (𝐹 ∪ 𝐻))
45	44	eleq1d 2823	. . . 4 ⊢ (𝑖 = 𝐹 → ((𝑖 ∪ 𝐻) ∈ 𝐼 ↔ (𝐹 ∪ 𝐻) ∈ 𝐼))
46	43, 45	anbi12d 631	. . 3 ⊢ (𝑖 = 𝐹 → ((𝐹 ≈ 𝑖 ∧ (𝑖 ∪ 𝐻) ∈ 𝐼) ↔ (𝐹 ≈ 𝐹 ∧ (𝐹 ∪ 𝐻) ∈ 𝐼)))
47	46	rspcev 3561	. 2 ⊢ ((𝐹 ∈ 𝒫 𝐺 ∧ (𝐹 ≈ 𝐹 ∧ (𝐹 ∪ 𝐻) ∈ 𝐼)) → ∃𝑖 ∈ 𝒫 𝐺(𝐹 ≈ 𝑖 ∧ (𝑖 ∪ 𝐻) ∈ 𝐼))
48	37, 42, 16, 47	syl12anc 834	1 ⊢ (𝜑 → ∃𝑖 ∈ 𝒫 𝐺(𝐹 ≈ 𝑖 ∧ (𝑖 ∪ 𝐻) ∈ 𝐼))

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ∖ cdif 3884   ∪ cun 3885   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  {csn 4561   class class class wbr 5074  ‘cfv 6433   ≈ cen 8730  Moorecmre 17291  mrClscmrc 17292  mrIndcmri 17293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-en 8734  df-mre 17295  df-mrc 17296  df-mri 17297

This theorem is referenced by:  mreexexlem4d  17356  mreexexd  17357