Theorem mreexexlem3d 17149

Description: Base case of the induction in mreexexd 17151. (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 488	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = ∅) → 𝐹 = ∅)
2		mreexexlem2d.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))
3	2	adantr 484	. . . . . . . . 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 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻)))
8		simpr 488	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐺 = ∅)
9	8	uneq1d 4076	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐺 = ∅) → (𝐺 ∪ 𝐻) = (∅ ∪ 𝐻))
10		uncom 4067	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∪ ∅) = (∅ ∪ 𝐻)
11		un0 4305	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∪ ∅) = 𝐻
12	10, 11	eqtr3i 2767	. . . . . . . . . . . . 13 ⊢ (∅ ∪ 𝐻) = 𝐻
13	9, 12	eqtrdi 2794	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐺 = ∅) → (𝐺 ∪ 𝐻) = 𝐻)
14	13	fveq2d 6721	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐺 = ∅) → (𝑁‘(𝐺 ∪ 𝐻)) = (𝑁‘𝐻))
15	7, 14	sseqtrd 3941	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐹 ⊆ (𝑁‘𝐻))
16		mreexexlem2d.8	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 ∪ 𝐻) ∈ 𝐼)
17	16	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐺 = ∅) → (𝐹 ∪ 𝐻) ∈ 𝐼)
18	5, 3, 17	mrissd 17139	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐺 = ∅) → (𝐹 ∪ 𝐻) ⊆ 𝑋)
19	18	unssbd 4102	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐻 ⊆ 𝑋)
20	3, 4, 19	mrcssidd 17128	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐻 ⊆ (𝑁‘𝐻))
21	15, 20	unssd 4100	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐺 = ∅) → (𝐹 ∪ 𝐻) ⊆ (𝑁‘𝐻))
22		ssun2 4087	. . . . . . . . . 10 ⊢ 𝐻 ⊆ (𝐹 ∪ 𝐻)
23	22	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐻 ⊆ (𝐹 ∪ 𝐻))
24	3, 4, 5, 21, 23, 17	mrissmrcd 17143	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐺 = ∅) → (𝐹 ∪ 𝐻) = 𝐻)
25		ssequn1 4094	. . . . . . . 8 ⊢ (𝐹 ⊆ 𝐻 ↔ (𝐹 ∪ 𝐻) = 𝐻)
26	24, 25	sylibr 237	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐹 ⊆ 𝐻)
27		mreexexlem2d.5	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ⊆ (𝑋 ∖ 𝐻))
28	27	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐹 ⊆ (𝑋 ∖ 𝐻))
29	26, 28	ssind 4147	. . . . . 6 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐹 ⊆ (𝐻 ∩ (𝑋 ∖ 𝐻)))
30		disjdif 4386	. . . . . 6 ⊢ (𝐻 ∩ (𝑋 ∖ 𝐻)) = ∅
31	29, 30	sseqtrdi 3951	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐹 ⊆ ∅)
32		ss0b 4312	. . . . 5 ⊢ (𝐹 ⊆ ∅ ↔ 𝐹 = ∅)
33	31, 32	sylib 221	. . . 4 ⊢ ((𝜑 ∧ 𝐺 = ∅) → 𝐹 = ∅)
34		mreexexlem3d.9	. . . 4 ⊢ (𝜑 → (𝐹 = ∅ ∨ 𝐺 = ∅))
35	1, 33, 34	mpjaodan 959	. . 3 ⊢ (𝜑 → 𝐹 = ∅)
36		0elpw 5247	. . 3 ⊢ ∅ ∈ 𝒫 𝐺
37	35, 36	eqeltrdi 2846	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝒫 𝐺)
38	2	elfvexd 6751	. . . 4 ⊢ (𝜑 → 𝑋 ∈ V)
39	27	difss2d 4049	. . . 4 ⊢ (𝜑 → 𝐹 ⊆ 𝑋)
40	38, 39	ssexd 5217	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
41		enrefg 8660	. . 3 ⊢ (𝐹 ∈ V → 𝐹 ≈ 𝐹)
42	40, 41	syl 17	. 2 ⊢ (𝜑 → 𝐹 ≈ 𝐹)
43		breq2 5057	. . . 4 ⊢ (𝑖 = 𝐹 → (𝐹 ≈ 𝑖 ↔ 𝐹 ≈ 𝐹))
44		uneq1 4070	. . . . 5 ⊢ (𝑖 = 𝐹 → (𝑖 ∪ 𝐻) = (𝐹 ∪ 𝐻))
45	44	eleq1d 2822	. . . 4 ⊢ (𝑖 = 𝐹 → ((𝑖 ∪ 𝐻) ∈ 𝐼 ↔ (𝐹 ∪ 𝐻) ∈ 𝐼))
46	43, 45	anbi12d 634	. . 3 ⊢ (𝑖 = 𝐹 → ((𝐹 ≈ 𝑖 ∧ (𝑖 ∪ 𝐻) ∈ 𝐼) ↔ (𝐹 ≈ 𝐹 ∧ (𝐹 ∪ 𝐻) ∈ 𝐼)))
47	46	rspcev 3537	. 2 ⊢ ((𝐹 ∈ 𝒫 𝐺 ∧ (𝐹 ≈ 𝐹 ∧ (𝐹 ∪ 𝐻) ∈ 𝐼)) → ∃𝑖 ∈ 𝒫 𝐺(𝐹 ≈ 𝑖 ∧ (𝑖 ∪ 𝐻) ∈ 𝐼))
48	37, 42, 16, 47	syl12anc 837	1 ⊢ (𝜑 → ∃𝑖 ∈ 𝒫 𝐺(𝐹 ≈ 𝑖 ∧ (𝑖 ∪ 𝐻) ∈ 𝐼))

Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 847   = wceq 1543   ∈ wcel 2110  ∀wral 3061  ∃wrex 3062  Vcvv 3408   ∖ cdif 3863   ∪ cun 3864   ∩ cin 3865   ⊆ wss 3866  ∅c0 4237  𝒫 cpw 4513  {csn 4541   class class class wbr 5053  ‘cfv 6380   ≈ cen 8623  Moorecmre 17085  mrClscmrc 17086  mrIndcmri 17087

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-int 4860  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-en 8627  df-mre 17089  df-mrc 17090  df-mri 17091

This theorem is referenced by:  mreexexlem4d  17150  mreexexd  17151