Theorem addccan2nclem2 6265

Description: Lemma for addccan2nc 6266. Establish stratification for induction. (Contributed by Scott Fenton, 2-Aug-2019.)

Assertion

Ref	Expression
addccan2nclem2	⊢ ((N ∈ V ∧ P ∈ W) → {x ∣ ((x +c N) = (x +c P) → N = P)} ∈ V)

Distinct variable groups:   x,N   x,P

Allowed substitution hints:   V(x)   W(x)

Proof of Theorem addccan2nclem2

Dummy variables n p y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		unab 3522	. . 3 ⊢ ({x ∣ ¬ (x +c N) = (x +c P)} ∪ {x ∣ N = P}) = {x ∣ (¬ (x +c N) = (x +c P) ∨ N = P)}
2		complab 3525	. . . 4 ⊢ ∼ {x ∣ (x +c N) = (x +c P)} = {x ∣ ¬ (x +c N) = (x +c P)}
3	2	uneq1i 3415	. . 3 ⊢ ( ∼ {x ∣ (x +c N) = (x +c P)} ∪ {x ∣ N = P}) = ({x ∣ ¬ (x +c N) = (x +c P)} ∪ {x ∣ N = P})
4		imor 401	. . . 4 ⊢ (((x +c N) = (x +c P) → N = P) ↔ (¬ (x +c N) = (x +c P) ∨ N = P))
5	4	abbii 2466	. . 3 ⊢ {x ∣ ((x +c N) = (x +c P) → N = P)} = {x ∣ (¬ (x +c N) = (x +c P) ∨ N = P)}
6	1, 3, 5	3eqtr4i 2383	. 2 ⊢ ( ∼ {x ∣ (x +c N) = (x +c P)} ∪ {x ∣ N = P}) = {x ∣ ((x +c N) = (x +c P) → N = P)}
7		addceq2 4385	. . . . . . . 8 ⊢ (n = N → (x +c n) = (x +c N))
8	7	eqeq1d 2361	. . . . . . 7 ⊢ (n = N → ((x +c n) = (x +c p) ↔ (x +c N) = (x +c p)))
9	8	abbidv 2468	. . . . . 6 ⊢ (n = N → {x ∣ (x +c n) = (x +c p)} = {x ∣ (x +c N) = (x +c p)})
10	9	eleq1d 2419	. . . . 5 ⊢ (n = N → ({x ∣ (x +c n) = (x +c p)} ∈ V ↔ {x ∣ (x +c N) = (x +c p)} ∈ V))
11		addceq2 4385	. . . . . . . 8 ⊢ (p = P → (x +c p) = (x +c P))
12	11	eqeq2d 2364	. . . . . . 7 ⊢ (p = P → ((x +c N) = (x +c p) ↔ (x +c N) = (x +c P)))
13	12	abbidv 2468	. . . . . 6 ⊢ (p = P → {x ∣ (x +c N) = (x +c p)} = {x ∣ (x +c N) = (x +c P)})
14	13	eleq1d 2419	. . . . 5 ⊢ (p = P → ({x ∣ (x +c N) = (x +c p)} ∈ V ↔ {x ∣ (x +c N) = (x +c P)} ∈ V))
15		elfix 5788	. . . . . . . 8 ⊢ (x ∈ Fix (◡( AddC ∘ ◡(1st ↾ (V × {p}))) ∘ ( AddC ∘ ◡(1st ↾ (V × {n})))) ↔ x(◡( AddC ∘ ◡(1st ↾ (V × {p}))) ∘ ( AddC ∘ ◡(1st ↾ (V × {n}))))x)
16		brco 4884	. . . . . . . . 9 ⊢ (x(◡( AddC ∘ ◡(1st ↾ (V × {p}))) ∘ ( AddC ∘ ◡(1st ↾ (V × {n}))))x ↔ ∃y(x( AddC ∘ ◡(1st ↾ (V × {n})))y ∧ y◡( AddC ∘ ◡(1st ↾ (V × {p})))x))
17		addccan2nclem1 6264	. . . . . . . . . . 11 ⊢ (x( AddC ∘ ◡(1st ↾ (V × {n})))y ↔ y = (x +c n))
18		brcnv 4893	. . . . . . . . . . . 12 ⊢ (y◡( AddC ∘ ◡(1st ↾ (V × {p})))x ↔ x( AddC ∘ ◡(1st ↾ (V × {p})))y)
19		addccan2nclem1 6264	. . . . . . . . . . . 12 ⊢ (x( AddC ∘ ◡(1st ↾ (V × {p})))y ↔ y = (x +c p))
20	18, 19	bitri 240	. . . . . . . . . . 11 ⊢ (y◡( AddC ∘ ◡(1st ↾ (V × {p})))x ↔ y = (x +c p))
21	17, 20	anbi12i 678	. . . . . . . . . 10 ⊢ ((x( AddC ∘ ◡(1st ↾ (V × {n})))y ∧ y◡( AddC ∘ ◡(1st ↾ (V × {p})))x) ↔ (y = (x +c n) ∧ y = (x +c p)))
22	21	exbii 1582	. . . . . . . . 9 ⊢ (∃y(x( AddC ∘ ◡(1st ↾ (V × {n})))y ∧ y◡( AddC ∘ ◡(1st ↾ (V × {p})))x) ↔ ∃y(y = (x +c n) ∧ y = (x +c p)))
23	16, 22	bitri 240	. . . . . . . 8 ⊢ (x(◡( AddC ∘ ◡(1st ↾ (V × {p}))) ∘ ( AddC ∘ ◡(1st ↾ (V × {n}))))x ↔ ∃y(y = (x +c n) ∧ y = (x +c p)))
24		vex 2863	. . . . . . . . . 10 ⊢ x ∈ V
25		vex 2863	. . . . . . . . . 10 ⊢ n ∈ V
26	24, 25	addcex 4395	. . . . . . . . 9 ⊢ (x +c n) ∈ V
27		eqeq1 2359	. . . . . . . . 9 ⊢ (y = (x +c n) → (y = (x +c p) ↔ (x +c n) = (x +c p)))
28	26, 27	ceqsexv 2895	. . . . . . . 8 ⊢ (∃y(y = (x +c n) ∧ y = (x +c p)) ↔ (x +c n) = (x +c p))
29	15, 23, 28	3bitri 262	. . . . . . 7 ⊢ (x ∈ Fix (◡( AddC ∘ ◡(1st ↾ (V × {p}))) ∘ ( AddC ∘ ◡(1st ↾ (V × {n})))) ↔ (x +c n) = (x +c p))
30	29	abbi2i 2465	. . . . . 6 ⊢ Fix (◡( AddC ∘ ◡(1st ↾ (V × {p}))) ∘ ( AddC ∘ ◡(1st ↾ (V × {n})))) = {x ∣ (x +c n) = (x +c p)}
31		addcfnex 5825	. . . . . . . . . 10 ⊢ AddC ∈ V
32		1stex 4740	. . . . . . . . . . . 12 ⊢ 1st ∈ V
33		vvex 4110	. . . . . . . . . . . . 13 ⊢ V ∈ V
34		snex 4112	. . . . . . . . . . . . 13 ⊢ {p} ∈ V
35	33, 34	xpex 5116	. . . . . . . . . . . 12 ⊢ (V × {p}) ∈ V
36	32, 35	resex 5118	. . . . . . . . . . 11 ⊢ (1st ↾ (V × {p})) ∈ V
37	36	cnvex 5103	. . . . . . . . . 10 ⊢ ◡(1st ↾ (V × {p})) ∈ V
38	31, 37	coex 4751	. . . . . . . . 9 ⊢ ( AddC ∘ ◡(1st ↾ (V × {p}))) ∈ V
39	38	cnvex 5103	. . . . . . . 8 ⊢ ◡( AddC ∘ ◡(1st ↾ (V × {p}))) ∈ V
40		snex 4112	. . . . . . . . . . . 12 ⊢ {n} ∈ V
41	33, 40	xpex 5116	. . . . . . . . . . 11 ⊢ (V × {n}) ∈ V
42	32, 41	resex 5118	. . . . . . . . . 10 ⊢ (1st ↾ (V × {n})) ∈ V
43	42	cnvex 5103	. . . . . . . . 9 ⊢ ◡(1st ↾ (V × {n})) ∈ V
44	31, 43	coex 4751	. . . . . . . 8 ⊢ ( AddC ∘ ◡(1st ↾ (V × {n}))) ∈ V
45	39, 44	coex 4751	. . . . . . 7 ⊢ (◡( AddC ∘ ◡(1st ↾ (V × {p}))) ∘ ( AddC ∘ ◡(1st ↾ (V × {n})))) ∈ V
46	45	fixex 5790	. . . . . 6 ⊢ Fix (◡( AddC ∘ ◡(1st ↾ (V × {p}))) ∘ ( AddC ∘ ◡(1st ↾ (V × {n})))) ∈ V
47	30, 46	eqeltrri 2424	. . . . 5 ⊢ {x ∣ (x +c n) = (x +c p)} ∈ V
48	10, 14, 47	vtocl2g 2919	. . . 4 ⊢ ((N ∈ V ∧ P ∈ W) → {x ∣ (x +c N) = (x +c P)} ∈ V)
49		complexg 4100	. . . 4 ⊢ ({x ∣ (x +c N) = (x +c P)} ∈ V → ∼ {x ∣ (x +c N) = (x +c P)} ∈ V)
50	48, 49	syl 15	. . 3 ⊢ ((N ∈ V ∧ P ∈ W) → ∼ {x ∣ (x +c N) = (x +c P)} ∈ V)
51		abexv 4325	. . 3 ⊢ {x ∣ N = P} ∈ V
52		unexg 4102	. . 3 ⊢ (( ∼ {x ∣ (x +c N) = (x +c P)} ∈ V ∧ {x ∣ N = P} ∈ V) → ( ∼ {x ∣ (x +c N) = (x +c P)} ∪ {x ∣ N = P}) ∈ V)
53	50, 51, 52	sylancl 643	. 2 ⊢ ((N ∈ V ∧ P ∈ W) → ( ∼ {x ∣ (x +c N) = (x +c P)} ∪ {x ∣ N = P}) ∈ V)
54	6, 53	syl5eqelr 2438	1 ⊢ ((N ∈ V ∧ P ∈ W) → {x ∣ ((x +c N) = (x +c P) → N = P)} ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2860   ∼ ccompl 3206   ∪ cun 3208  {csn 3738   +_c cplc 4376   class class class wbr 4640  1^st c1st 4718   ∘ ccom 4722   × cxp 4771  ^◡ccnv 4772   ↾ cres 4775   Fix cfix 5740   AddC caddcfn 5746

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-13 1712  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 theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-iun 3972  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-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-fo 4794  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-fix 5741  df-cup 5743  df-disj 5745  df-addcfn 5747  df-ins2 5751  df-ins3 5753  df-ins4 5757  df-si3 5759

This theorem is referenced by:  addccan2nc  6266