Theorem peano3nninf 12882

Description: The successor function on ℕ_∞ is never zero. Half of Lemma 3.4 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 1-Aug-2022.)

Hypothesis

Ref	Expression
peano3nninf.s	|- S = ( p e. ℕ∞ |-> ( i e. om |-> if ( i = (/) , 1o , ( p ` U. i ) ) ) )

Assertion

Ref	Expression
peano3nninf	|- ( A e. ℕ∞ -> ( S ` A ) =/= ( x e. om |-> (/) ) )

Distinct variable groups:   A, i, p   S, i, x   x, p

Allowed substitution hints:   A( x)   S( p)

Proof of Theorem peano3nninf

Step	Hyp	Ref	Expression
1		fveq1 5372	. . . . . . . . . 10 |- ( p = A -> ( p ` U. i ) = ( A ` U. i ) )
2	1	ifeq2d 3454	. . . . . . . . 9 |- ( p = A -> if ( i = (/) , 1o , ( p ` U. i ) ) = if ( i = (/) , 1o , ( A ` U. i ) ) )
3	2	mpteq2dv 3977	. . . . . . . 8 |- ( p = A -> ( i e. om |-> if ( i = (/) , 1o , ( p ` U. i ) ) ) = ( i e. om |-> if ( i = (/) , 1o , ( A ` U. i ) ) ) )
4		peano3nninf.s	. . . . . . . 8 |- S = ( p e. ℕ∞ |-> ( i e. om |-> if ( i = (/) , 1o , ( p ` U. i ) ) ) )
5		omex 4465	. . . . . . . . 9 |- om e. _V
6	5	mptex 5598	. . . . . . . 8 |- ( i e. om |-> if ( i = (/) , 1o , ( A ` U. i ) ) ) e. _V
7	3, 4, 6	fvmpt 5450	. . . . . . 7 |- ( A e. ℕ∞ -> ( S ` A ) = ( i e. om |-> if ( i = (/) , 1o , ( A ` U. i ) ) ) )
8		eqeq1 2119	. . . . . . . . 9 |- ( i = (/) -> ( i = (/) <-> (/) = (/) ) )
9		unieq 3709	. . . . . . . . . 10 |- ( i = (/) -> U. i = U. (/) )
10	9	fveq2d 5377	. . . . . . . . 9 |- ( i = (/) -> ( A ` U. i ) = ( A ` U. (/) ) )
11	8, 10	ifbieq2d 3460	. . . . . . . 8 |- ( i = (/) -> if ( i = (/) , 1o , ( A ` U. i ) ) = if ( (/) = (/) , 1o , ( A ` U. (/) ) ) )
12	11	adantl 273	. . . . . . 7 |- ( ( A e. ℕ∞ /\ i = (/) ) -> if ( i = (/) , 1o , ( A ` U. i ) ) = if ( (/) = (/) , 1o , ( A ` U. (/) ) ) )
13		peano1 4466	. . . . . . . 8 |- (/) e. om
14	13	a1i 9	. . . . . . 7 |- ( A e. ℕ∞ -> (/) e. om )
15		eqid 2113	. . . . . . . . . 10 |- (/) = (/)
16	15	iftruei 3444	. . . . . . . . 9 |- if ( (/) = (/) , 1o , ( A ` U. (/) ) ) = 1o
17		1onn 6368	. . . . . . . . 9 |- 1o e. om
18	16, 17	eqeltri 2185	. . . . . . . 8 |- if ( (/) = (/) , 1o , ( A ` U. (/) ) ) e. om
19	18	a1i 9	. . . . . . 7 |- ( A e. ℕ∞ -> if ( (/) = (/) , 1o , ( A ` U. (/) ) ) e. om )
20	7, 12, 14, 19	fvmptd 5454	. . . . . 6 |- ( A e. ℕ∞ -> ( ( S ` A ) ` (/) ) = if ( (/) = (/) , 1o , ( A ` U. (/) ) ) )
21	20, 16	syl6eq 2161	. . . . 5 |- ( A e. ℕ∞ -> ( ( S ` A ) ` (/) ) = 1o )
22	21	adantr 272	. . . 4 |- ( ( A e. ℕ∞ /\ ( S ` A ) = ( x e. om |-> (/) ) ) -> ( ( S ` A ) ` (/) ) = 1o )
23		fveq1 5372	. . . . . 6 |- ( ( S ` A ) = ( x e. om |-> (/) ) -> ( ( S ` A ) ` (/) ) = ( ( x e. om |-> (/) ) ` (/) ) )
24	23	adantl 273	. . . . 5 |- ( ( A e. ℕ∞ /\ ( S ` A ) = ( x e. om |-> (/) ) ) -> ( ( S ` A ) ` (/) ) = ( ( x e. om |-> (/) ) ` (/) ) )
25	15	a1i 9	. . . . . . 7 |- ( x = (/) -> (/) = (/) )
26		eqid 2113	. . . . . . 7 |- ( x e. om |-> (/) ) = ( x e. om |-> (/) )
27	25, 26	fvmptg 5449	. . . . . 6 |- ( ( (/) e. om /\ (/) e. om ) -> ( ( x e. om |-> (/) ) ` (/) ) = (/) )
28	13, 13, 27	mp2an 420	. . . . 5 |- ( ( x e. om |-> (/) ) ` (/) ) = (/)
29	24, 28	syl6eq 2161	. . . 4 |- ( ( A e. ℕ∞ /\ ( S ` A ) = ( x e. om |-> (/) ) ) -> ( ( S ` A ) ` (/) ) = (/) )
30	22, 29	eqtr3d 2147	. . 3 |- ( ( A e. ℕ∞ /\ ( S ` A ) = ( x e. om |-> (/) ) ) -> 1o = (/) )
31		1n0 6281	. . . . 5 |- 1o =/= (/)
32	31	neii 2282	. . . 4 |- -. 1o = (/)
33	32	a1i 9	. . 3 |- ( ( A e. ℕ∞ /\ ( S ` A ) = ( x e. om |-> (/) ) ) -> -. 1o = (/) )
34	30, 33	pm2.65da 633	. 2 |- ( A e. ℕ∞ -> -. ( S ` A ) = ( x e. om |-> (/) ) )
35	34	neqned 2287	1 |- ( A e. ℕ∞ -> ( S ` A ) =/= ( x e. om |-> (/) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1312   e. wcel 1461   =/= wne 2280   (/)c0 3327   ifcif 3438   U.cuni 3700   |-> cmpt 3947   omcom 4462   ` cfv 5079   1oc1o 6258  ℕ_∞xnninf 6953

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-iinf 4460

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-if 3439  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-1o 6265

This theorem is referenced by:  exmidsbthrlem  12898