Theorem nnadjoinpw 4522

Description: Adjoining an element to a power class. Theorem X.1.40 of [Rosser] p. 530. (Contributed by SF, 27-Jan-2015.)

Assertion

Ref	Expression
nnadjoinpw	⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (A ∈ M ∧ X ∈ ∼ A) ∧ 𝒫 A ∈ N) → 𝒫 (A ∪ {X}) ∈ (N +c N))

Proof of Theorem nnadjoinpw

Dummy variables a b t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwadjoin 4120	. 2 ⊢ 𝒫 (A ∪ {X}) = (𝒫 A ∪ {a ∣ ∃b ∈ 𝒫 Aa = (b ∪ {X})})
2		simp3 957	. . 3 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (A ∈ M ∧ X ∈ ∼ A) ∧ 𝒫 A ∈ N) → 𝒫 A ∈ N)
3		simp1r 980	. . . 4 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (A ∈ M ∧ X ∈ ∼ A) ∧ 𝒫 A ∈ N) → N ∈ Nn )
4		simp2r 982	. . . . 5 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (A ∈ M ∧ X ∈ ∼ A) ∧ 𝒫 A ∈ N) → X ∈ ∼ A)
5		unipw 4118	. . . . . 6 ⊢ ∪𝒫 A = A
6	5	compleqi 3245	. . . . 5 ⊢ ∼ ∪𝒫 A = ∼ A
7	4, 6	syl6eleqr 2444	. . . 4 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (A ∈ M ∧ X ∈ ∼ A) ∧ 𝒫 A ∈ N) → X ∈ ∼ ∪𝒫 A)
8		nnadjoin 4521	. . . 4 ⊢ ((N ∈ Nn ∧ 𝒫 A ∈ N ∧ X ∈ ∼ ∪𝒫 A) → {a ∣ ∃b ∈ 𝒫 Aa = (b ∪ {X})} ∈ N)
9	3, 2, 7, 8	syl3anc 1182	. . 3 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (A ∈ M ∧ X ∈ ∼ A) ∧ 𝒫 A ∈ N) → {a ∣ ∃b ∈ 𝒫 Aa = (b ∪ {X})} ∈ N)
10		elcomplg 3219	. . . . . . . . 9 ⊢ (X ∈ ∼ A → (X ∈ ∼ A ↔ ¬ X ∈ A))
11	10	ibi 232	. . . . . . . 8 ⊢ (X ∈ ∼ A → ¬ X ∈ A)
12	4, 11	syl 15	. . . . . . 7 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (A ∈ M ∧ X ∈ ∼ A) ∧ 𝒫 A ∈ N) → ¬ X ∈ A)
13		snssg 3845	. . . . . . . 8 ⊢ (X ∈ ∼ A → (X ∈ A ↔ {X} ⊆ A))
14	4, 13	syl 15	. . . . . . 7 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (A ∈ M ∧ X ∈ ∼ A) ∧ 𝒫 A ∈ N) → (X ∈ A ↔ {X} ⊆ A))
15	12, 14	mtbid 291	. . . . . 6 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (A ∈ M ∧ X ∈ ∼ A) ∧ 𝒫 A ∈ N) → ¬ {X} ⊆ A)
16	15	intnand 882	. . . . 5 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (A ∈ M ∧ X ∈ ∼ A) ∧ 𝒫 A ∈ N) → ¬ (b ⊆ A ∧ {X} ⊆ A))
17	16	ralrimivw 2699	. . . 4 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (A ∈ M ∧ X ∈ ∼ A) ∧ 𝒫 A ∈ N) → ∀b ∈ 𝒫 A ¬ (b ⊆ A ∧ {X} ⊆ A))
18		disjr 3593	. . . . 5 ⊢ ((𝒫 A ∩ {a ∣ ∃b ∈ 𝒫 Aa = (b ∪ {X})}) = ∅ ↔ ∀t ∈ {a ∣ ∃b ∈ 𝒫 Aa = (b ∪ {X})} ¬ t ∈ 𝒫 A)
19		eqeq1 2359	. . . . . . 7 ⊢ (a = t → (a = (b ∪ {X}) ↔ t = (b ∪ {X})))
20	19	rexbidv 2636	. . . . . 6 ⊢ (a = t → (∃b ∈ 𝒫 Aa = (b ∪ {X}) ↔ ∃b ∈ 𝒫 At = (b ∪ {X})))
21	20	ralab 2998	. . . . 5 ⊢ (∀t ∈ {a ∣ ∃b ∈ 𝒫 Aa = (b ∪ {X})} ¬ t ∈ 𝒫 A ↔ ∀t(∃b ∈ 𝒫 At = (b ∪ {X}) → ¬ t ∈ 𝒫 A))
22		ralcom4 2878	. . . . . 6 ⊢ (∀b ∈ 𝒫 A∀t(t = (b ∪ {X}) → ¬ t ∈ 𝒫 A) ↔ ∀t∀b ∈ 𝒫 A(t = (b ∪ {X}) → ¬ t ∈ 𝒫 A))
23		vex 2863	. . . . . . . . . 10 ⊢ b ∈ V
24		snex 4112	. . . . . . . . . 10 ⊢ {X} ∈ V
25	23, 24	unex 4107	. . . . . . . . 9 ⊢ (b ∪ {X}) ∈ V
26		eleq1 2413	. . . . . . . . . 10 ⊢ (t = (b ∪ {X}) → (t ∈ 𝒫 A ↔ (b ∪ {X}) ∈ 𝒫 A))
27	26	notbid 285	. . . . . . . . 9 ⊢ (t = (b ∪ {X}) → (¬ t ∈ 𝒫 A ↔ ¬ (b ∪ {X}) ∈ 𝒫 A))
28	25, 27	ceqsalv 2886	. . . . . . . 8 ⊢ (∀t(t = (b ∪ {X}) → ¬ t ∈ 𝒫 A) ↔ ¬ (b ∪ {X}) ∈ 𝒫 A)
29	25	elpw 3729	. . . . . . . . 9 ⊢ ((b ∪ {X}) ∈ 𝒫 A ↔ (b ∪ {X}) ⊆ A)
30		unss 3438	. . . . . . . . 9 ⊢ ((b ⊆ A ∧ {X} ⊆ A) ↔ (b ∪ {X}) ⊆ A)
31	29, 30	bitr4i 243	. . . . . . . 8 ⊢ ((b ∪ {X}) ∈ 𝒫 A ↔ (b ⊆ A ∧ {X} ⊆ A))
32	28, 31	xchbinx 301	. . . . . . 7 ⊢ (∀t(t = (b ∪ {X}) → ¬ t ∈ 𝒫 A) ↔ ¬ (b ⊆ A ∧ {X} ⊆ A))
33	32	ralbii 2639	. . . . . 6 ⊢ (∀b ∈ 𝒫 A∀t(t = (b ∪ {X}) → ¬ t ∈ 𝒫 A) ↔ ∀b ∈ 𝒫 A ¬ (b ⊆ A ∧ {X} ⊆ A))
34		r19.23v 2731	. . . . . . 7 ⊢ (∀b ∈ 𝒫 A(t = (b ∪ {X}) → ¬ t ∈ 𝒫 A) ↔ (∃b ∈ 𝒫 At = (b ∪ {X}) → ¬ t ∈ 𝒫 A))
35	34	albii 1566	. . . . . 6 ⊢ (∀t∀b ∈ 𝒫 A(t = (b ∪ {X}) → ¬ t ∈ 𝒫 A) ↔ ∀t(∃b ∈ 𝒫 At = (b ∪ {X}) → ¬ t ∈ 𝒫 A))
36	22, 33, 35	3bitr3ri 267	. . . . 5 ⊢ (∀t(∃b ∈ 𝒫 At = (b ∪ {X}) → ¬ t ∈ 𝒫 A) ↔ ∀b ∈ 𝒫 A ¬ (b ⊆ A ∧ {X} ⊆ A))
37	18, 21, 36	3bitri 262	. . . 4 ⊢ ((𝒫 A ∩ {a ∣ ∃b ∈ 𝒫 Aa = (b ∪ {X})}) = ∅ ↔ ∀b ∈ 𝒫 A ¬ (b ⊆ A ∧ {X} ⊆ A))
38	17, 37	sylibr 203	. . 3 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (A ∈ M ∧ X ∈ ∼ A) ∧ 𝒫 A ∈ N) → (𝒫 A ∩ {a ∣ ∃b ∈ 𝒫 Aa = (b ∪ {X})}) = ∅)
39		eladdci 4400	. . 3 ⊢ ((𝒫 A ∈ N ∧ {a ∣ ∃b ∈ 𝒫 Aa = (b ∪ {X})} ∈ N ∧ (𝒫 A ∩ {a ∣ ∃b ∈ 𝒫 Aa = (b ∪ {X})}) = ∅) → (𝒫 A ∪ {a ∣ ∃b ∈ 𝒫 Aa = (b ∪ {X})}) ∈ (N +c N))
40	2, 9, 38, 39	syl3anc 1182	. 2 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (A ∈ M ∧ X ∈ ∼ A) ∧ 𝒫 A ∈ N) → (𝒫 A ∪ {a ∣ ∃b ∈ 𝒫 Aa = (b ∪ {X})}) ∈ (N +c N))
41	1, 40	syl5eqel 2437	1 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (A ∈ M ∧ X ∈ ∼ A) ∧ 𝒫 A ∈ N) → 𝒫 (A ∪ {X}) ∈ (N +c N))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2615  ∃wrex 2616   ∼ ccompl 3206   ∪ cun 3208   ∩ cin 3209   ⊆ wss 3258  ∅c0 3551  𝒫 cpw 3723  {csn 3738  ∪cuni 3892   Nn cnnc 4374   +_c cplc 4376

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-0c 4378  df-addc 4379  df-nnc 4380

This theorem is used by:  nnpweq  4524  sfindbl  4531