Theorem nnadjoinpw 4521

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) ∧ ℘A ∈ N) → ℘(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 4119	. 2 ⊢ ℘(A ∪ {X}) = (℘A ∪ {a ∣ ∃b ∈ ℘ Aa = (b ∪ {X})})
2		simp3 957	. . 3 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (A ∈ M ∧ X ∈ ∼ A) ∧ ℘A ∈ N) → ℘A ∈ N)
3		simp1r 980	. . . 4 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (A ∈ M ∧ X ∈ ∼ A) ∧ ℘A ∈ N) → N ∈ Nn )
4		simp2r 982	. . . . 5 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (A ∈ M ∧ X ∈ ∼ A) ∧ ℘A ∈ N) → X ∈ ∼ A)
5		unipw 4117	. . . . . 6 ⊢ ∪℘A = A
6	5	compleqi 3244	. . . . 5 ⊢ ∼ ∪℘A = ∼ A
7	4, 6	syl6eleqr 2444	. . . 4 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (A ∈ M ∧ X ∈ ∼ A) ∧ ℘A ∈ N) → X ∈ ∼ ∪℘A)
8		nnadjoin 4520	. . . 4 ⊢ ((N ∈ Nn ∧ ℘A ∈ N ∧ X ∈ ∼ ∪℘A) → {a ∣ ∃b ∈ ℘ Aa = (b ∪ {X})} ∈ N)
9	3, 2, 7, 8	syl3anc 1182	. . 3 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (A ∈ M ∧ X ∈ ∼ A) ∧ ℘A ∈ N) → {a ∣ ∃b ∈ ℘ Aa = (b ∪ {X})} ∈ N)
10		elcomplg 3218	. . . . . . . . 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) ∧ ℘A ∈ N) → ¬ X ∈ A)
13		snssg 3844	. . . . . . . 8 ⊢ (X ∈ ∼ A → (X ∈ A ↔ {X} ⊆ A))
14	4, 13	syl 15	. . . . . . 7 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (A ∈ M ∧ X ∈ ∼ A) ∧ ℘A ∈ N) → (X ∈ A ↔ {X} ⊆ A))
15	12, 14	mtbid 291	. . . . . 6 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (A ∈ M ∧ X ∈ ∼ A) ∧ ℘A ∈ N) → ¬ {X} ⊆ A)
16	15	intnand 882	. . . . 5 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (A ∈ M ∧ X ∈ ∼ A) ∧ ℘A ∈ N) → ¬ (b ⊆ A ∧ {X} ⊆ A))
17	16	ralrimivw 2698	. . . 4 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (A ∈ M ∧ X ∈ ∼ A) ∧ ℘A ∈ N) → ∀b ∈ ℘ A ¬ (b ⊆ A ∧ {X} ⊆ A))
18		disjr 3592	. . . . 5 ⊢ ((℘A ∩ {a ∣ ∃b ∈ ℘ Aa = (b ∪ {X})}) = ∅ ↔ ∀t ∈ {a ∣ ∃b ∈ ℘ Aa = (b ∪ {X})} ¬ t ∈ ℘A)
19		eqeq1 2359	. . . . . . 7 ⊢ (a = t → (a = (b ∪ {X}) ↔ t = (b ∪ {X})))
20	19	rexbidv 2635	. . . . . 6 ⊢ (a = t → (∃b ∈ ℘ Aa = (b ∪ {X}) ↔ ∃b ∈ ℘ At = (b ∪ {X})))
21	20	ralab 2997	. . . . 5 ⊢ (∀t ∈ {a ∣ ∃b ∈ ℘ Aa = (b ∪ {X})} ¬ t ∈ ℘A ↔ ∀t(∃b ∈ ℘ At = (b ∪ {X}) → ¬ t ∈ ℘A))
22		ralcom4 2877	. . . . . 6 ⊢ (∀b ∈ ℘ A∀t(t = (b ∪ {X}) → ¬ t ∈ ℘A) ↔ ∀t∀b ∈ ℘ A(t = (b ∪ {X}) → ¬ t ∈ ℘A))
23		vex 2862	. . . . . . . . . 10 ⊢ b ∈ V
24		snex 4111	. . . . . . . . . 10 ⊢ {X} ∈ V
25	23, 24	unex 4106	. . . . . . . . 9 ⊢ (b ∪ {X}) ∈ V
26		eleq1 2413	. . . . . . . . . 10 ⊢ (t = (b ∪ {X}) → (t ∈ ℘A ↔ (b ∪ {X}) ∈ ℘A))
27	26	notbid 285	. . . . . . . . 9 ⊢ (t = (b ∪ {X}) → (¬ t ∈ ℘A ↔ ¬ (b ∪ {X}) ∈ ℘A))
28	25, 27	ceqsalv 2885	. . . . . . . 8 ⊢ (∀t(t = (b ∪ {X}) → ¬ t ∈ ℘A) ↔ ¬ (b ∪ {X}) ∈ ℘A)
29	25	elpw 3728	. . . . . . . . 9 ⊢ ((b ∪ {X}) ∈ ℘A ↔ (b ∪ {X}) ⊆ A)
30		unss 3437	. . . . . . . . 9 ⊢ ((b ⊆ A ∧ {X} ⊆ A) ↔ (b ∪ {X}) ⊆ A)
31	29, 30	bitr4i 243	. . . . . . . 8 ⊢ ((b ∪ {X}) ∈ ℘A ↔ (b ⊆ A ∧ {X} ⊆ A))
32	28, 31	xchbinx 301	. . . . . . 7 ⊢ (∀t(t = (b ∪ {X}) → ¬ t ∈ ℘A) ↔ ¬ (b ⊆ A ∧ {X} ⊆ A))
33	32	ralbii 2638	. . . . . 6 ⊢ (∀b ∈ ℘ A∀t(t = (b ∪ {X}) → ¬ t ∈ ℘A) ↔ ∀b ∈ ℘ A ¬ (b ⊆ A ∧ {X} ⊆ A))
34		r19.23v 2730	. . . . . . 7 ⊢ (∀b ∈ ℘ A(t = (b ∪ {X}) → ¬ t ∈ ℘A) ↔ (∃b ∈ ℘ At = (b ∪ {X}) → ¬ t ∈ ℘A))
35	34	albii 1566	. . . . . 6 ⊢ (∀t∀b ∈ ℘ A(t = (b ∪ {X}) → ¬ t ∈ ℘A) ↔ ∀t(∃b ∈ ℘ At = (b ∪ {X}) → ¬ t ∈ ℘A))
36	22, 33, 35	3bitr3ri 267	. . . . 5 ⊢ (∀t(∃b ∈ ℘ At = (b ∪ {X}) → ¬ t ∈ ℘A) ↔ ∀b ∈ ℘ A ¬ (b ⊆ A ∧ {X} ⊆ A))
37	18, 21, 36	3bitri 262	. . . 4 ⊢ ((℘A ∩ {a ∣ ∃b ∈ ℘ Aa = (b ∪ {X})}) = ∅ ↔ ∀b ∈ ℘ A ¬ (b ⊆ A ∧ {X} ⊆ A))
38	17, 37	sylibr 203	. . 3 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (A ∈ M ∧ X ∈ ∼ A) ∧ ℘A ∈ N) → (℘A ∩ {a ∣ ∃b ∈ ℘ Aa = (b ∪ {X})}) = ∅)
39		eladdci 4399	. . 3 ⊢ ((℘A ∈ N ∧ {a ∣ ∃b ∈ ℘ Aa = (b ∪ {X})} ∈ N ∧ (℘A ∩ {a ∣ ∃b ∈ ℘ Aa = (b ∪ {X})}) = ∅) → (℘A ∪ {a ∣ ∃b ∈ ℘ Aa = (b ∪ {X})}) ∈ (N +c N))
40	2, 9, 38, 39	syl3anc 1182	. 2 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (A ∈ M ∧ X ∈ ∼ A) ∧ ℘A ∈ N) → (℘A ∪ {a ∣ ∃b ∈ ℘ Aa = (b ∪ {X})}) ∈ (N +c N))
41	1, 40	syl5eqel 2437	1 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (A ∈ M ∧ X ∈ ∼ A) ∧ ℘A ∈ N) → ℘(A ∪ {X}) ∈ (N +c N))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2614  ∃wrex 2615   ∼ ccompl 3205   ∪ cun 3207   ∩ cin 3208   ⊆ wss 3257  ∅c0 3550  ℘cpw 3722  {csn 3737  ∪cuni 3891   Nn cnnc 4373   +_c cplc 4375

This theorem was proved from 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-0c 4377  df-addc 4378  df-nnc 4379

This theorem is referenced by:  nnpweq  4523  sfindbl  4530