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Theorem peano4nc 6150
 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.
StepHypRef Expression
1 peano2nc 6145 . . . . . 6 (A NC → (A +c 1c) NC )
21adantr 451 . . . . 5 ((A NC B NC ) → (A +c 1c) NC )
32adantr 451 . . . 4 (((A NC B NC ) (A +c 1c) = (B +c 1c)) → (A +c 1c) NC )
4 elncs 6119 . . . . 5 ((A +c 1c) NCg(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)
75, 6jca 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 2862 . . . . . . . . . . . . . 14 g V
98ncid 6123 . . . . . . . . . . . . 13 g Nc g
10 eleq2 2414 . . . . . . . . . . . . 13 ((A +c 1c) = Nc g → (g (A +c 1c) ↔ g Nc g))
119, 10mpbiri 224 . . . . . . . . . . . 12 ((A +c 1c) = Nc gg (A +c 1c))
12 elsuc 4413 . . . . . . . . . . . . 13 (g (A +c 1c) ↔ t A x tg = (t ∪ {x}))
1312biimpi 186 . . . . . . . . . . . 12 (g (A +c 1c) → t A x tg = (t ∪ {x}))
1411, 13syl 15 . . . . . . . . . . 11 ((A +c 1c) = Nc gt A x tg = (t ∪ {x}))
15 eleq2 2414 . . . . . . . . . . . . 13 ((B +c 1c) = Nc g → (g (B +c 1c) ↔ g Nc g))
169, 15mpbiri 224 . . . . . . . . . . . 12 ((B +c 1c) = Nc gg (B +c 1c))
17 elsuc 4413 . . . . . . . . . . . 12 (g (B +c 1c) ↔ f B y fg = (f ∪ {y}))
1816, 17sylib 188 . . . . . . . . . . 11 ((B +c 1c) = Nc gf B y fg = (f ∪ {y}))
1914, 18anim12i 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 2778 . . . . . . . . . . . . 13 (x ty f(g = (t ∪ {x}) g = (f ∪ {y})) ↔ (x tg = (t ∪ {x}) y fg = (f ∪ {y})))
21202rexbii 2641 . . . . . . . . . . . 12 (t A f B x ty f(g = (t ∪ {x}) g = (f ∪ {y})) ↔ t A f B (x tg = (t ∪ {x}) y fg = (f ∪ {y})))
22 reeanv 2778 . . . . . . . . . . . 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})))
2321, 22bitri 240 . . . . . . . . . . 11 (t A f B x ty f(g = (t ∪ {x}) g = (f ∪ {y})) ↔ (t A x tg = (t ∪ {x}) f B y fg = (f ∪ {y})))
24 ncseqnc 6128 . . . . . . . . . . . . . . 15 (A NC → (A = Nc tt A))
25 ncseqnc 6128 . . . . . . . . . . . . . . 15 (B NC → (B = Nc ff B))
2624, 25bi2anan9 843 . . . . . . . . . . . . . 14 ((A NC B NC ) → ((A = Nc t B = Nc f) ↔ (t A f B)))
2726biimpar 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 2862 . . . . . . . . . . . . . . . . . 18 x V
3029elcompl 3225 . . . . . . . . . . . . . . . . 17 (x t ↔ ¬ x t)
31 vex 2862 . . . . . . . . . . . . . . . . . 18 y V
3231elcompl 3225 . . . . . . . . . . . . . . . . 17 (y f ↔ ¬ y f)
33 vex 2862 . . . . . . . . . . . . . . . . . . . . 21 t V
34 vex 2862 . . . . . . . . . . . . . . . . . . . . 21 f V
3533, 34, 29, 31enadj 6060 . . . . . . . . . . . . . . . . . . . 20 (((t ∪ {x}) = (f ∪ {y}) ¬ x t ¬ y f) → tf)
36353expb 1152 . . . . . . . . . . . . . . . . . . 19 (((t ∪ {x}) = (f ∪ {y}) x t ¬ y f)) → tf)
3736ancoms 439 . . . . . . . . . . . . . . . . . 18 (((¬ x t ¬ y f) (t ∪ {x}) = (f ∪ {y})) → tf)
3837ex 423 . . . . . . . . . . . . . . . . 17 ((¬ x t ¬ y f) → ((t ∪ {x}) = (f ∪ {y}) → tf))
3930, 32, 38syl2anb 465 . . . . . . . . . . . . . . . 16 ((x t y f) → ((t ∪ {x}) = (f ∪ {y}) → tf))
4028, 39syl5 28 . . . . . . . . . . . . . . 15 ((x t y f) → ((g = (t ∪ {x}) g = (f ∪ {y})) → tf))
4140rexlimivv 2743 . . . . . . . . . . . . . 14 (x ty f(g = (t ∪ {x}) g = (f ∪ {y})) → tf)
42 eqeq12 2365 . . . . . . . . . . . . . . 15 ((A = Nc t B = Nc f) → (A = BNc t = Nc f))
4333eqnc 6127 . . . . . . . . . . . . . . 15 ( Nc t = Nc ftf)
4442, 43syl6bb 252 . . . . . . . . . . . . . 14 ((A = Nc t B = Nc f) → (A = Btf))
4541, 44syl5ibr 212 . . . . . . . . . . . . 13 ((A = Nc t B = Nc f) → (x ty f(g = (t ∪ {x}) g = (f ∪ {y})) → A = B))
4627, 45syl 15 . . . . . . . . . . . 12 (((A NC B NC ) (t A f B)) → (x ty f(g = (t ∪ {x}) g = (f ∪ {y})) → A = B))
4746rexlimdvva 2745 . . . . . . . . . . 11 ((A NC B NC ) → (t A f B x ty f(g = (t ∪ {x}) g = (f ∪ {y})) → A = B))
4823, 47syl5bir 209 . . . . . . . . . 10 ((A NC B NC ) → ((t A x tg = (t ∪ {x}) f B y fg = (f ∪ {y})) → A = B))
4919, 48syl5 28 . . . . . . . . 9 ((A NC B NC ) → (((A +c 1c) = Nc g (B +c 1c) = Nc g) → A = B))
5049imp 418 . . . . . . . 8 (((A NC B NC ) ((A +c 1c) = Nc g (B +c 1c) = Nc g)) → A = B)
517, 50sylan2 460 . . . . . . 7 (((A NC B NC ) ((A +c 1c) = (B +c 1c) (A +c 1c) = Nc g)) → A = B)
5251expr 598 . . . . . 6 (((A NC B NC ) (A +c 1c) = (B +c 1c)) → ((A +c 1c) = Nc gA = B))
5352exlimdv 1636 . . . . 5 (((A NC B NC ) (A +c 1c) = (B +c 1c)) → (g(A +c 1c) = Nc gA = B))
544, 53syl5bi 208 . . . 4 (((A NC B NC ) (A +c 1c) = (B +c 1c)) → ((A +c 1c) NCA = B))
553, 54mpd 14 . . 3 (((A NC B NC ) (A +c 1c) = (B +c 1c)) → A = B)
5655ex 423 . 2 ((A NC B NC ) → ((A +c 1c) = (B +c 1c) → A = B))
57 addceq1 4383 . 2 (A = B → (A +c 1c) = (B +c 1c))
5856, 57impbid1 194 1 ((A NC B NC ) → ((A +c 1c) = (B +c 1c) ↔ A = B))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   ∼ ccompl 3205   ∪ cun 3207  {csn 3737  1cc1c 4134   +c cplc 4375   class class class wbr 4639   ≈ cen 6028   NC cncs 6088   Nc cnc 6091 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 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-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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  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-pss 3261  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-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-2nd 4797  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-en 6029  df-ncs 6098  df-nc 6101 This theorem is referenced by:  nclenn  6249  addccan2nc  6265  ncslesuc  6267  nchoicelem12  6300  nchoicelem14  6302  nchoicelem17  6305
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