Theorem peano4nc 6151

Description: Successor is one-to-one over the cardinals. Theorem XI.2.12 of [Rosser] p. 375. (Contributed by SF, 25-Feb-2015.)

Assertion

Ref	Expression
peano4nc	⊢ ((A ∈ NC ∧ B ∈ NC ) → ((A +c 1c) = (B +c 1c) ↔ A = B))

Proof of Theorem peano4nc

Dummy variables f g t x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano2nc 6146	. . . . . 6 ⊢ (A ∈ NC → (A +c 1c) ∈ NC )
2	1	adantr 451	. . . . 5 ⊢ ((A ∈ NC ∧ B ∈ NC ) → (A +c 1c) ∈ NC )
3	2	adantr 451	. . . 4 ⊢ (((A ∈ NC ∧ B ∈ NC ) ∧ (A +c 1c) = (B +c 1c)) → (A +c 1c) ∈ NC )
4		elncs 6120	. . . . 5 ⊢ ((A +c 1c) ∈ NC ↔ ∃g(A +c 1c) = Nc g)
5		simpr 447	. . . . . . . . 9 ⊢ (((A +c 1c) = (B +c 1c) ∧ (A +c 1c) = Nc g) → (A +c 1c) = Nc g)
6		eqtr2 2371	. . . . . . . . 9 ⊢ (((A +c 1c) = (B +c 1c) ∧ (A +c 1c) = Nc g) → (B +c 1c) = Nc g)
7	5, 6	jca 518	. . . . . . . 8 ⊢ (((A +c 1c) = (B +c 1c) ∧ (A +c 1c) = Nc g) → ((A +c 1c) = Nc g ∧ (B +c 1c) = Nc g))
8		vex 2863	. . . . . . . . . . . . . 14 ⊢ g ∈ V
9	8	ncid 6124	. . . . . . . . . . . . 13 ⊢ g ∈ Nc g
10		eleq2 2414	. . . . . . . . . . . . 13 ⊢ ((A +c 1c) = Nc g → (g ∈ (A +c 1c) ↔ g ∈ Nc g))
11	9, 10	mpbiri 224	. . . . . . . . . . . 12 ⊢ ((A +c 1c) = Nc g → g ∈ (A +c 1c))
12		elsuc 4414	. . . . . . . . . . . . 13 ⊢ (g ∈ (A +c 1c) ↔ ∃t ∈ A ∃x ∈ ∼ tg = (t ∪ {x}))
13	12	biimpi 186	. . . . . . . . . . . 12 ⊢ (g ∈ (A +c 1c) → ∃t ∈ A ∃x ∈ ∼ tg = (t ∪ {x}))
14	11, 13	syl 15	. . . . . . . . . . 11 ⊢ ((A +c 1c) = Nc g → ∃t ∈ A ∃x ∈ ∼ tg = (t ∪ {x}))
15		eleq2 2414	. . . . . . . . . . . . 13 ⊢ ((B +c 1c) = Nc g → (g ∈ (B +c 1c) ↔ g ∈ Nc g))
16	9, 15	mpbiri 224	. . . . . . . . . . . 12 ⊢ ((B +c 1c) = Nc g → g ∈ (B +c 1c))
17		elsuc 4414	. . . . . . . . . . . 12 ⊢ (g ∈ (B +c 1c) ↔ ∃f ∈ B ∃y ∈ ∼ fg = (f ∪ {y}))
18	16, 17	sylib 188	. . . . . . . . . . 11 ⊢ ((B +c 1c) = Nc g → ∃f ∈ B ∃y ∈ ∼ fg = (f ∪ {y}))
19	14, 18	anim12i 549	. . . . . . . . . 10 ⊢ (((A +c 1c) = Nc g ∧ (B +c 1c) = Nc g) → (∃t ∈ A ∃x ∈ ∼ tg = (t ∪ {x}) ∧ ∃f ∈ B ∃y ∈ ∼ fg = (f ∪ {y})))
20		reeanv 2779	. . . . . . . . . . . . 13 ⊢ (∃x ∈ ∼ t∃y ∈ ∼ f(g = (t ∪ {x}) ∧ g = (f ∪ {y})) ↔ (∃x ∈ ∼ tg = (t ∪ {x}) ∧ ∃y ∈ ∼ fg = (f ∪ {y})))
21	20	2rexbii 2642	. . . . . . . . . . . 12 ⊢ (∃t ∈ A ∃f ∈ B ∃x ∈ ∼ t∃y ∈ ∼ f(g = (t ∪ {x}) ∧ g = (f ∪ {y})) ↔ ∃t ∈ A ∃f ∈ B (∃x ∈ ∼ tg = (t ∪ {x}) ∧ ∃y ∈ ∼ fg = (f ∪ {y})))
22		reeanv 2779	. . . . . . . . . . . 12 ⊢ (∃t ∈ A ∃f ∈ B (∃x ∈ ∼ tg = (t ∪ {x}) ∧ ∃y ∈ ∼ fg = (f ∪ {y})) ↔ (∃t ∈ A ∃x ∈ ∼ tg = (t ∪ {x}) ∧ ∃f ∈ B ∃y ∈ ∼ fg = (f ∪ {y})))
23	21, 22	bitri 240	. . . . . . . . . . 11 ⊢ (∃t ∈ A ∃f ∈ B ∃x ∈ ∼ t∃y ∈ ∼ f(g = (t ∪ {x}) ∧ g = (f ∪ {y})) ↔ (∃t ∈ A ∃x ∈ ∼ tg = (t ∪ {x}) ∧ ∃f ∈ B ∃y ∈ ∼ fg = (f ∪ {y})))
24		ncseqnc 6129	. . . . . . . . . . . . . . 15 ⊢ (A ∈ NC → (A = Nc t ↔ t ∈ A))
25		ncseqnc 6129	. . . . . . . . . . . . . . 15 ⊢ (B ∈ NC → (B = Nc f ↔ f ∈ B))
26	24, 25	bi2anan9 843	. . . . . . . . . . . . . 14 ⊢ ((A ∈ NC ∧ B ∈ NC ) → ((A = Nc t ∧ B = Nc f) ↔ (t ∈ A ∧ f ∈ B)))
27	26	biimpar 471	. . . . . . . . . . . . 13 ⊢ (((A ∈ NC ∧ B ∈ NC ) ∧ (t ∈ A ∧ f ∈ B)) → (A = Nc t ∧ B = Nc f))
28		eqtr2 2371	. . . . . . . . . . . . . . . 16 ⊢ ((g = (t ∪ {x}) ∧ g = (f ∪ {y})) → (t ∪ {x}) = (f ∪ {y}))
29		vex 2863	. . . . . . . . . . . . . . . . . 18 ⊢ x ∈ V
30	29	elcompl 3226	. . . . . . . . . . . . . . . . 17 ⊢ (x ∈ ∼ t ↔ ¬ x ∈ t)
31		vex 2863	. . . . . . . . . . . . . . . . . 18 ⊢ y ∈ V
32	31	elcompl 3226	. . . . . . . . . . . . . . . . 17 ⊢ (y ∈ ∼ f ↔ ¬ y ∈ f)
33		vex 2863	. . . . . . . . . . . . . . . . . . . . 21 ⊢ t ∈ V
34		vex 2863	. . . . . . . . . . . . . . . . . . . . 21 ⊢ f ∈ V
35	33, 34, 29, 31	enadj 6061	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((t ∪ {x}) = (f ∪ {y}) ∧ ¬ x ∈ t ∧ ¬ y ∈ f) → t ≈ f)
36	35	3expb 1152	. . . . . . . . . . . . . . . . . . 19 ⊢ (((t ∪ {x}) = (f ∪ {y}) ∧ (¬ x ∈ t ∧ ¬ y ∈ f)) → t ≈ f)
37	36	ancoms 439	. . . . . . . . . . . . . . . . . 18 ⊢ (((¬ x ∈ t ∧ ¬ y ∈ f) ∧ (t ∪ {x}) = (f ∪ {y})) → t ≈ f)
38	37	ex 423	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ x ∈ t ∧ ¬ y ∈ f) → ((t ∪ {x}) = (f ∪ {y}) → t ≈ f))
39	30, 32, 38	syl2anb 465	. . . . . . . . . . . . . . . 16 ⊢ ((x ∈ ∼ t ∧ y ∈ ∼ f) → ((t ∪ {x}) = (f ∪ {y}) → t ≈ f))
40	28, 39	syl5 28	. . . . . . . . . . . . . . 15 ⊢ ((x ∈ ∼ t ∧ y ∈ ∼ f) → ((g = (t ∪ {x}) ∧ g = (f ∪ {y})) → t ≈ f))
41	40	rexlimivv 2744	. . . . . . . . . . . . . 14 ⊢ (∃x ∈ ∼ t∃y ∈ ∼ f(g = (t ∪ {x}) ∧ g = (f ∪ {y})) → t ≈ f)
42		eqeq12 2365	. . . . . . . . . . . . . . 15 ⊢ ((A = Nc t ∧ B = Nc f) → (A = B ↔ Nc t = Nc f))
43	33	eqnc 6128	. . . . . . . . . . . . . . 15 ⊢ ( Nc t = Nc f ↔ t ≈ f)
44	42, 43	syl6bb 252	. . . . . . . . . . . . . 14 ⊢ ((A = Nc t ∧ B = Nc f) → (A = B ↔ t ≈ f))
45	41, 44	syl5ibr 212	. . . . . . . . . . . . 13 ⊢ ((A = Nc t ∧ B = Nc f) → (∃x ∈ ∼ t∃y ∈ ∼ f(g = (t ∪ {x}) ∧ g = (f ∪ {y})) → A = B))
46	27, 45	syl 15	. . . . . . . . . . . 12 ⊢ (((A ∈ NC ∧ B ∈ NC ) ∧ (t ∈ A ∧ f ∈ B)) → (∃x ∈ ∼ t∃y ∈ ∼ f(g = (t ∪ {x}) ∧ g = (f ∪ {y})) → A = B))
47	46	rexlimdvva 2746	. . . . . . . . . . 11 ⊢ ((A ∈ NC ∧ B ∈ NC ) → (∃t ∈ A ∃f ∈ B ∃x ∈ ∼ t∃y ∈ ∼ f(g = (t ∪ {x}) ∧ g = (f ∪ {y})) → A = B))
48	23, 47	syl5bir 209	. . . . . . . . . 10 ⊢ ((A ∈ NC ∧ B ∈ NC ) → ((∃t ∈ A ∃x ∈ ∼ tg = (t ∪ {x}) ∧ ∃f ∈ B ∃y ∈ ∼ fg = (f ∪ {y})) → A = B))
49	19, 48	syl5 28	. . . . . . . . 9 ⊢ ((A ∈ NC ∧ B ∈ NC ) → (((A +c 1c) = Nc g ∧ (B +c 1c) = Nc g) → A = B))
50	49	imp 418	. . . . . . . 8 ⊢ (((A ∈ NC ∧ B ∈ NC ) ∧ ((A +c 1c) = Nc g ∧ (B +c 1c) = Nc g)) → A = B)
51	7, 50	sylan2 460	. . . . . . 7 ⊢ (((A ∈ NC ∧ B ∈ NC ) ∧ ((A +c 1c) = (B +c 1c) ∧ (A +c 1c) = Nc g)) → A = B)
52	51	expr 598	. . . . . 6 ⊢ (((A ∈ NC ∧ B ∈ NC ) ∧ (A +c 1c) = (B +c 1c)) → ((A +c 1c) = Nc g → A = B))
53	52	exlimdv 1636	. . . . 5 ⊢ (((A ∈ NC ∧ B ∈ NC ) ∧ (A +c 1c) = (B +c 1c)) → (∃g(A +c 1c) = Nc g → A = B))
54	4, 53	syl5bi 208	. . . 4 ⊢ (((A ∈ NC ∧ B ∈ NC ) ∧ (A +c 1c) = (B +c 1c)) → ((A +c 1c) ∈ NC → A = B))
55	3, 54	mpd 14	. . 3 ⊢ (((A ∈ NC ∧ B ∈ NC ) ∧ (A +c 1c) = (B +c 1c)) → A = B)
56	55	ex 423	. 2 ⊢ ((A ∈ NC ∧ B ∈ NC ) → ((A +c 1c) = (B +c 1c) → A = B))
57		addceq1 4384	. 2 ⊢ (A = B → (A +c 1c) = (B +c 1c))
58	56, 57	impbid1 194	1 ⊢ ((A ∈ NC ∧ B ∈ NC ) → ((A +c 1c) = (B +c 1c) ↔ A = B))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616   ∼ ccompl 3206   ∪ cun 3208  {csn 3738  1_cc1c 4135   +_c cplc 4376   class class class wbr 4640   ≈ cen 6029   NC cncs 6089   Nc cnc 6092

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-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-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-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-txp 5737  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-en 6030  df-ncs 6099  df-nc 6102

This theorem is referenced by:  nclenn  6250  addccan2nc  6266  ncslesuc  6268  nchoicelem12  6301  nchoicelem14  6303  nchoicelem17  6306