Theorem prepeano4 4452

Description: Assuming a non-null successor, cardinal successor is one-to-one. Theorem X.1.19 of [Rosser] p. 526. (Contributed by SF, 18-Jan-2015.)

Assertion

Ref	Expression
prepeano4	⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ ((M +c 1c) = (N +c 1c) ∧ (M +c 1c) ≠ ∅)) → M = N)

Proof of Theorem prepeano4

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

Step	Hyp	Ref	Expression
1		n0 3560	. . 3 ⊢ ((M +c 1c) ≠ ∅ ↔ ∃a a ∈ (M +c 1c))
2		elsuc 4414	. . . . 5 ⊢ (a ∈ (M +c 1c) ↔ ∃b ∈ M ∃x ∈ ∼ ba = (b ∪ {x}))
3		simplll 734	. . . . . . . 8 ⊢ ((((M ∈ Nn ∧ N ∈ Nn ) ∧ (M +c 1c) = (N +c 1c)) ∧ (b ∈ M ∧ x ∈ ∼ b)) → M ∈ Nn )
4		simpllr 735	. . . . . . . 8 ⊢ ((((M ∈ Nn ∧ N ∈ Nn ) ∧ (M +c 1c) = (N +c 1c)) ∧ (b ∈ M ∧ x ∈ ∼ b)) → N ∈ Nn )
5		simprl 732	. . . . . . . 8 ⊢ ((((M ∈ Nn ∧ N ∈ Nn ) ∧ (M +c 1c) = (N +c 1c)) ∧ (b ∈ M ∧ x ∈ ∼ b)) → b ∈ M)
6		simprr 733	. . . . . . . . . 10 ⊢ ((((M ∈ Nn ∧ N ∈ Nn ) ∧ (M +c 1c) = (N +c 1c)) ∧ (b ∈ M ∧ x ∈ ∼ b)) → x ∈ ∼ b)
7		vex 2863	. . . . . . . . . . 11 ⊢ x ∈ V
8	7	elcompl 3226	. . . . . . . . . 10 ⊢ (x ∈ ∼ b ↔ ¬ x ∈ b)
9	6, 8	sylib 188	. . . . . . . . 9 ⊢ ((((M ∈ Nn ∧ N ∈ Nn ) ∧ (M +c 1c) = (N +c 1c)) ∧ (b ∈ M ∧ x ∈ ∼ b)) → ¬ x ∈ b)
10	7	elsuci 4415	. . . . . . . . . . . 12 ⊢ ((b ∈ M ∧ ¬ x ∈ b) → (b ∪ {x}) ∈ (M +c 1c))
11	8, 10	sylan2b 461	. . . . . . . . . . 11 ⊢ ((b ∈ M ∧ x ∈ ∼ b) → (b ∪ {x}) ∈ (M +c 1c))
12	11	adantl 452	. . . . . . . . . 10 ⊢ ((((M ∈ Nn ∧ N ∈ Nn ) ∧ (M +c 1c) = (N +c 1c)) ∧ (b ∈ M ∧ x ∈ ∼ b)) → (b ∪ {x}) ∈ (M +c 1c))
13		simplr 731	. . . . . . . . . 10 ⊢ ((((M ∈ Nn ∧ N ∈ Nn ) ∧ (M +c 1c) = (N +c 1c)) ∧ (b ∈ M ∧ x ∈ ∼ b)) → (M +c 1c) = (N +c 1c))
14	12, 13	eleqtrd 2429	. . . . . . . . 9 ⊢ ((((M ∈ Nn ∧ N ∈ Nn ) ∧ (M +c 1c) = (N +c 1c)) ∧ (b ∈ M ∧ x ∈ ∼ b)) → (b ∪ {x}) ∈ (N +c 1c))
15		vex 2863	. . . . . . . . . 10 ⊢ b ∈ V
16	15, 7	nnsucelr 4429	. . . . . . . . 9 ⊢ ((N ∈ Nn ∧ (¬ x ∈ b ∧ (b ∪ {x}) ∈ (N +c 1c))) → b ∈ N)
17	4, 9, 14, 16	syl12anc 1180	. . . . . . . 8 ⊢ ((((M ∈ Nn ∧ N ∈ Nn ) ∧ (M +c 1c) = (N +c 1c)) ∧ (b ∈ M ∧ x ∈ ∼ b)) → b ∈ N)
18		nnceleq 4431	. . . . . . . 8 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (b ∈ M ∧ b ∈ N)) → M = N)
19	3, 4, 5, 17, 18	syl22anc 1183	. . . . . . 7 ⊢ ((((M ∈ Nn ∧ N ∈ Nn ) ∧ (M +c 1c) = (N +c 1c)) ∧ (b ∈ M ∧ x ∈ ∼ b)) → M = N)
20	19	a1d 22	. . . . . 6 ⊢ ((((M ∈ Nn ∧ N ∈ Nn ) ∧ (M +c 1c) = (N +c 1c)) ∧ (b ∈ M ∧ x ∈ ∼ b)) → (a = (b ∪ {x}) → M = N))
21	20	rexlimdvva 2746	. . . . 5 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (M +c 1c) = (N +c 1c)) → (∃b ∈ M ∃x ∈ ∼ ba = (b ∪ {x}) → M = N))
22	2, 21	syl5bi 208	. . . 4 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (M +c 1c) = (N +c 1c)) → (a ∈ (M +c 1c) → M = N))
23	22	exlimdv 1636	. . 3 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (M +c 1c) = (N +c 1c)) → (∃a a ∈ (M +c 1c) → M = N))
24	1, 23	syl5bi 208	. 2 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (M +c 1c) = (N +c 1c)) → ((M +c 1c) ≠ ∅ → M = N))
25	24	impr 602	1 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ ((M +c 1c) = (N +c 1c) ∧ (M +c 1c) ≠ ∅)) → M = N)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  ∃wrex 2616   ∼ ccompl 3206   ∪ cun 3208  ∅c0 3551  {csn 3738  1_cc1c 4135   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-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-idk 4196  df-0c 4378  df-addc 4379  df-nnc 4380

This theorem is used by:  preaddccan2  4456  evenodddisj  4517  peano4  4558