Theorem peano3nninf 13201

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 5420	. . . . . . . . . 10 |- ( p = A -> ( p ` U. i ) = ( A ` U. i ) )
2	1	ifeq2d 3490	. . . . . . . . 9 |- ( p = A -> if ( i = (/) , 1o , ( p ` U. i ) ) = if ( i = (/) , 1o , ( A ` U. i ) ) )
3	2	mpteq2dv 4019	. . . . . . . 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 4507	. . . . . . . . 9 |- om e. _V
6	5	mptex 5646	. . . . . . . 8 |- ( i e. om |-> if ( i = (/) , 1o , ( A ` U. i ) ) ) e. _V
7	3, 4, 6	fvmpt 5498	. . . . . . 7 |- ( A e. ℕ∞ -> ( S ` A ) = ( i e. om |-> if ( i = (/) , 1o , ( A ` U. i ) ) ) )
8		eqeq1 2146	. . . . . . . . 9 |- ( i = (/) -> ( i = (/) <-> (/) = (/) ) )
9		unieq 3745	. . . . . . . . . 10 |- ( i = (/) -> U. i = U. (/) )
10	9	fveq2d 5425	. . . . . . . . 9 |- ( i = (/) -> ( A ` U. i ) = ( A ` U. (/) ) )
11	8, 10	ifbieq2d 3496	. . . . . . . 8 |- ( i = (/) -> if ( i = (/) , 1o , ( A ` U. i ) ) = if ( (/) = (/) , 1o , ( A ` U. (/) ) ) )
12	11	adantl 275	. . . . . . 7 |- ( ( A e. ℕ∞ /\ i = (/) ) -> if ( i = (/) , 1o , ( A ` U. i ) ) = if ( (/) = (/) , 1o , ( A ` U. (/) ) ) )
13		peano1 4508	. . . . . . . 8 |- (/) e. om
14	13	a1i 9	. . . . . . 7 |- ( A e. ℕ∞ -> (/) e. om )
15		eqid 2139	. . . . . . . . . 10 |- (/) = (/)
16	15	iftruei 3480	. . . . . . . . 9 |- if ( (/) = (/) , 1o , ( A ` U. (/) ) ) = 1o
17		1onn 6416	. . . . . . . . 9 |- 1o e. om
18	16, 17	eqeltri 2212	. . . . . . . 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 5502	. . . . . 6 |- ( A e. ℕ∞ -> ( ( S ` A ) ` (/) ) = if ( (/) = (/) , 1o , ( A ` U. (/) ) ) )
21	20, 16	syl6eq 2188	. . . . 5 |- ( A e. ℕ∞ -> ( ( S ` A ) ` (/) ) = 1o )
22	21	adantr 274	. . . 4 |- ( ( A e. ℕ∞ /\ ( S ` A ) = ( x e. om |-> (/) ) ) -> ( ( S ` A ) ` (/) ) = 1o )
23		fveq1 5420	. . . . . 6 |- ( ( S ` A ) = ( x e. om |-> (/) ) -> ( ( S ` A ) ` (/) ) = ( ( x e. om |-> (/) ) ` (/) ) )
24	23	adantl 275	. . . . 5 |- ( ( A e. ℕ∞ /\ ( S ` A ) = ( x e. om |-> (/) ) ) -> ( ( S ` A ) ` (/) ) = ( ( x e. om |-> (/) ) ` (/) ) )
25	15	a1i 9	. . . . . . 7 |- ( x = (/) -> (/) = (/) )
26		eqid 2139	. . . . . . 7 |- ( x e. om |-> (/) ) = ( x e. om |-> (/) )
27	25, 26	fvmptg 5497	. . . . . 6 |- ( ( (/) e. om /\ (/) e. om ) -> ( ( x e. om |-> (/) ) ` (/) ) = (/) )
28	13, 13, 27	mp2an 422	. . . . 5 |- ( ( x e. om |-> (/) ) ` (/) ) = (/)
29	24, 28	syl6eq 2188	. . . 4 |- ( ( A e. ℕ∞ /\ ( S ` A ) = ( x e. om |-> (/) ) ) -> ( ( S ` A ) ` (/) ) = (/) )
30	22, 29	eqtr3d 2174	. . 3 |- ( ( A e. ℕ∞ /\ ( S ` A ) = ( x e. om |-> (/) ) ) -> 1o = (/) )
31		1n0 6329	. . . . 5 |- 1o =/= (/)
32	31	neii 2310	. . . 4 |- -. 1o = (/)
33	32	a1i 9	. . 3 |- ( ( A e. ℕ∞ /\ ( S ` A ) = ( x e. om |-> (/) ) ) -> -. 1o = (/) )
34	30, 33	pm2.65da 650	. 2 |- ( A e. ℕ∞ -> -. ( S ` A ) = ( x e. om |-> (/) ) )
35	34	neqned 2315	1 |- ( A e. ℕ∞ -> ( S ` A ) =/= ( x e. om |-> (/) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   =/= wne 2308   (/)c0 3363   ifcif 3474   U.cuni 3736   |-> cmpt 3989   omcom 4504   ` cfv 5123   1oc1o 6306  ℕ_∞xnninf 7005

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1o 6313

This theorem is referenced by:  exmidsbthrlem  13217