Theorem peano5 4410

Description: The principle of mathematical induction: a set containing cardinal zero and closed under the successor operator is a superset of the finite cardinals. Theorem X.1.6 of [Rosser] p. 276. (Contributed by SF, 14-Jan-2015.)

Assertion

Ref	Expression
peano5	⊢ ((A ∈ V ∧ 0c ∈ A ∧ ∀x ∈ Nn (x ∈ A → (x +c 1c) ∈ A)) → Nn ⊆ A)

Distinct variable group:   x,A

Allowed substitution hint:   V(x)

Proof of Theorem peano5

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nncex 4397	. . 3 ⊢ Nn ∈ V
2		inexg 4101	. . 3 ⊢ (( Nn ∈ V ∧ A ∈ V) → ( Nn ∩ A) ∈ V)
3	1, 2	mpan 651	. 2 ⊢ (A ∈ V → ( Nn ∩ A) ∈ V)
4		peano1 4403	. . 3 ⊢ 0c ∈ Nn
5		elin 3220	. . . 4 ⊢ (0c ∈ ( Nn ∩ A) ↔ (0c ∈ Nn ∧ 0c ∈ A))
6	5	biimpri 197	. . 3 ⊢ ((0c ∈ Nn ∧ 0c ∈ A) → 0c ∈ ( Nn ∩ A))
7	4, 6	mpan 651	. 2 ⊢ (0c ∈ A → 0c ∈ ( Nn ∩ A))
8		elin 3220	. . . . . 6 ⊢ (x ∈ ( Nn ∩ A) ↔ (x ∈ Nn ∧ x ∈ A))
9	8	imbi1i 315	. . . . 5 ⊢ ((x ∈ ( Nn ∩ A) → (x +c 1c) ∈ A) ↔ ((x ∈ Nn ∧ x ∈ A) → (x +c 1c) ∈ A))
10		impexp 433	. . . . 5 ⊢ (((x ∈ Nn ∧ x ∈ A) → (x +c 1c) ∈ A) ↔ (x ∈ Nn → (x ∈ A → (x +c 1c) ∈ A)))
11	9, 10	bitri 240	. . . 4 ⊢ ((x ∈ ( Nn ∩ A) → (x +c 1c) ∈ A) ↔ (x ∈ Nn → (x ∈ A → (x +c 1c) ∈ A)))
12		inss1 3476	. . . . . . . 8 ⊢ ( Nn ∩ A) ⊆ Nn
13	12	sseli 3270	. . . . . . 7 ⊢ (x ∈ ( Nn ∩ A) → x ∈ Nn )
14		peano2 4404	. . . . . . 7 ⊢ (x ∈ Nn → (x +c 1c) ∈ Nn )
15	13, 14	syl 15	. . . . . 6 ⊢ (x ∈ ( Nn ∩ A) → (x +c 1c) ∈ Nn )
16		elin 3220	. . . . . . . 8 ⊢ ((x +c 1c) ∈ ( Nn ∩ A) ↔ ((x +c 1c) ∈ Nn ∧ (x +c 1c) ∈ A))
17	16	biimpri 197	. . . . . . 7 ⊢ (((x +c 1c) ∈ Nn ∧ (x +c 1c) ∈ A) → (x +c 1c) ∈ ( Nn ∩ A))
18	17	a1i 10	. . . . . 6 ⊢ (x ∈ ( Nn ∩ A) → (((x +c 1c) ∈ Nn ∧ (x +c 1c) ∈ A) → (x +c 1c) ∈ ( Nn ∩ A)))
19	15, 18	mpand 656	. . . . 5 ⊢ (x ∈ ( Nn ∩ A) → ((x +c 1c) ∈ A → (x +c 1c) ∈ ( Nn ∩ A)))
20	19	a2i 12	. . . 4 ⊢ ((x ∈ ( Nn ∩ A) → (x +c 1c) ∈ A) → (x ∈ ( Nn ∩ A) → (x +c 1c) ∈ ( Nn ∩ A)))
21	11, 20	sylbir 204	. . 3 ⊢ ((x ∈ Nn → (x ∈ A → (x +c 1c) ∈ A)) → (x ∈ ( Nn ∩ A) → (x +c 1c) ∈ ( Nn ∩ A)))
22	21	ralimi2 2687	. 2 ⊢ (∀x ∈ Nn (x ∈ A → (x +c 1c) ∈ A) → ∀x ∈ ( Nn ∩ A)(x +c 1c) ∈ ( Nn ∩ A))
23		df-nnc 4380	. . . 4 ⊢ Nn = ∩{y ∣ (0c ∈ y ∧ ∀x ∈ y (x +c 1c) ∈ y)}
24		eleq2 2414	. . . . . . . . 9 ⊢ (y = ( Nn ∩ A) → (0c ∈ y ↔ 0c ∈ ( Nn ∩ A)))
25		eleq2 2414	. . . . . . . . . 10 ⊢ (y = ( Nn ∩ A) → ((x +c 1c) ∈ y ↔ (x +c 1c) ∈ ( Nn ∩ A)))
26	25	raleqbi1dv 2816	. . . . . . . . 9 ⊢ (y = ( Nn ∩ A) → (∀x ∈ y (x +c 1c) ∈ y ↔ ∀x ∈ ( Nn ∩ A)(x +c 1c) ∈ ( Nn ∩ A)))
27	24, 26	anbi12d 691	. . . . . . . 8 ⊢ (y = ( Nn ∩ A) → ((0c ∈ y ∧ ∀x ∈ y (x +c 1c) ∈ y) ↔ (0c ∈ ( Nn ∩ A) ∧ ∀x ∈ ( Nn ∩ A)(x +c 1c) ∈ ( Nn ∩ A))))
28	27	elabg 2987	. . . . . . 7 ⊢ (( Nn ∩ A) ∈ V → (( Nn ∩ A) ∈ {y ∣ (0c ∈ y ∧ ∀x ∈ y (x +c 1c) ∈ y)} ↔ (0c ∈ ( Nn ∩ A) ∧ ∀x ∈ ( Nn ∩ A)(x +c 1c) ∈ ( Nn ∩ A))))
29	28	biimprd 214	. . . . . 6 ⊢ (( Nn ∩ A) ∈ V → ((0c ∈ ( Nn ∩ A) ∧ ∀x ∈ ( Nn ∩ A)(x +c 1c) ∈ ( Nn ∩ A)) → ( Nn ∩ A) ∈ {y ∣ (0c ∈ y ∧ ∀x ∈ y (x +c 1c) ∈ y)}))
30	29	3impib 1149	. . . . 5 ⊢ ((( Nn ∩ A) ∈ V ∧ 0c ∈ ( Nn ∩ A) ∧ ∀x ∈ ( Nn ∩ A)(x +c 1c) ∈ ( Nn ∩ A)) → ( Nn ∩ A) ∈ {y ∣ (0c ∈ y ∧ ∀x ∈ y (x +c 1c) ∈ y)})
31		intss1 3942	. . . . 5 ⊢ (( Nn ∩ A) ∈ {y ∣ (0c ∈ y ∧ ∀x ∈ y (x +c 1c) ∈ y)} → ∩{y ∣ (0c ∈ y ∧ ∀x ∈ y (x +c 1c) ∈ y)} ⊆ ( Nn ∩ A))
32	30, 31	syl 15	. . . 4 ⊢ ((( Nn ∩ A) ∈ V ∧ 0c ∈ ( Nn ∩ A) ∧ ∀x ∈ ( Nn ∩ A)(x +c 1c) ∈ ( Nn ∩ A)) → ∩{y ∣ (0c ∈ y ∧ ∀x ∈ y (x +c 1c) ∈ y)} ⊆ ( Nn ∩ A))
33	23, 32	syl5eqss 3316	. . 3 ⊢ ((( Nn ∩ A) ∈ V ∧ 0c ∈ ( Nn ∩ A) ∧ ∀x ∈ ( Nn ∩ A)(x +c 1c) ∈ ( Nn ∩ A)) → Nn ⊆ ( Nn ∩ A))
34		inss2 3477	. . 3 ⊢ ( Nn ∩ A) ⊆ A
35	33, 34	syl6ss 3285	. 2 ⊢ ((( Nn ∩ A) ∈ V ∧ 0c ∈ ( Nn ∩ A) ∧ ∀x ∈ ( Nn ∩ A)(x +c 1c) ∈ ( Nn ∩ A)) → Nn ⊆ A)
36	3, 7, 22, 35	syl3an 1224	1 ⊢ ((A ∈ V ∧ 0c ∈ A ∧ ∀x ∈ Nn (x ∈ A → (x +c 1c) ∈ A)) → Nn ⊆ A)

Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2615  Vcvv 2860   ∩ cin 3209   ⊆ wss 3258  ∩cint 3927  1_cc1c 4135   Nn cnnc 4374  0_cc0c 4375   +_c cplc 4376

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-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-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-0c 4378  df-addc 4379  df-nnc 4380

This theorem is referenced by:  findsd  4411  dmfrec  6317