Theorem peano3nninf 15651

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 5557	. . . . . . . . . 10 |- ( p = A -> ( p ` U. i ) = ( A ` U. i ) )
2	1	ifeq2d 3579	. . . . . . . . 9 |- ( p = A -> if ( i = (/) , 1o , ( p ` U. i ) ) = if ( i = (/) , 1o , ( A ` U. i ) ) )
3	2	mpteq2dv 4124	. . . . . . . 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 4629	. . . . . . . . 9 |- om e. _V
6	5	mptex 5788	. . . . . . . 8 |- ( i e. om |-> if ( i = (/) , 1o , ( A ` U. i ) ) ) e. _V
7	3, 4, 6	fvmpt 5638	. . . . . . 7 |- ( A e. ℕ∞ -> ( S ` A ) = ( i e. om |-> if ( i = (/) , 1o , ( A ` U. i ) ) ) )
8		eqeq1 2203	. . . . . . . . 9 |- ( i = (/) -> ( i = (/) <-> (/) = (/) ) )
9		unieq 3848	. . . . . . . . . 10 |- ( i = (/) -> U. i = U. (/) )
10	9	fveq2d 5562	. . . . . . . . 9 |- ( i = (/) -> ( A ` U. i ) = ( A ` U. (/) ) )
11	8, 10	ifbieq2d 3585	. . . . . . . 8 |- ( i = (/) -> if ( i = (/) , 1o , ( A ` U. i ) ) = if ( (/) = (/) , 1o , ( A ` U. (/) ) ) )
12	11	adantl 277	. . . . . . 7 |- ( ( A e. ℕ∞ /\ i = (/) ) -> if ( i = (/) , 1o , ( A ` U. i ) ) = if ( (/) = (/) , 1o , ( A ` U. (/) ) ) )
13		peano1 4630	. . . . . . . 8 |- (/) e. om
14	13	a1i 9	. . . . . . 7 |- ( A e. ℕ∞ -> (/) e. om )
15		eqid 2196	. . . . . . . . . 10 |- (/) = (/)
16	15	iftruei 3567	. . . . . . . . 9 |- if ( (/) = (/) , 1o , ( A ` U. (/) ) ) = 1o
17		1onn 6578	. . . . . . . . 9 |- 1o e. om
18	16, 17	eqeltri 2269	. . . . . . . 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 5642	. . . . . 6 |- ( A e. ℕ∞ -> ( ( S ` A ) ` (/) ) = if ( (/) = (/) , 1o , ( A ` U. (/) ) ) )
21	20, 16	eqtrdi 2245	. . . . 5 |- ( A e. ℕ∞ -> ( ( S ` A ) ` (/) ) = 1o )
22	21	adantr 276	. . . 4 |- ( ( A e. ℕ∞ /\ ( S ` A ) = ( x e. om |-> (/) ) ) -> ( ( S ` A ) ` (/) ) = 1o )
23		fveq1 5557	. . . . . 6 |- ( ( S ` A ) = ( x e. om |-> (/) ) -> ( ( S ` A ) ` (/) ) = ( ( x e. om |-> (/) ) ` (/) ) )
24	23	adantl 277	. . . . 5 |- ( ( A e. ℕ∞ /\ ( S ` A ) = ( x e. om |-> (/) ) ) -> ( ( S ` A ) ` (/) ) = ( ( x e. om |-> (/) ) ` (/) ) )
25	15	a1i 9	. . . . . . 7 |- ( x = (/) -> (/) = (/) )
26		eqid 2196	. . . . . . 7 |- ( x e. om |-> (/) ) = ( x e. om |-> (/) )
27	25, 26	fvmptg 5637	. . . . . 6 |- ( ( (/) e. om /\ (/) e. om ) -> ( ( x e. om |-> (/) ) ` (/) ) = (/) )
28	13, 13, 27	mp2an 426	. . . . 5 |- ( ( x e. om |-> (/) ) ` (/) ) = (/)
29	24, 28	eqtrdi 2245	. . . 4 |- ( ( A e. ℕ∞ /\ ( S ` A ) = ( x e. om |-> (/) ) ) -> ( ( S ` A ) ` (/) ) = (/) )
30	22, 29	eqtr3d 2231	. . 3 |- ( ( A e. ℕ∞ /\ ( S ` A ) = ( x e. om |-> (/) ) ) -> 1o = (/) )
31		1n0 6490	. . . . 5 |- 1o =/= (/)
32	31	neii 2369	. . . 4 |- -. 1o = (/)
33	32	a1i 9	. . 3 |- ( ( A e. ℕ∞ /\ ( S ` A ) = ( x e. om |-> (/) ) ) -> -. 1o = (/) )
34	30, 33	pm2.65da 662	. 2 |- ( A e. ℕ∞ -> -. ( S ` A ) = ( x e. om |-> (/) ) )
35	34	neqned 2374	1 |- ( A e. ℕ∞ -> ( S ` A ) =/= ( x e. om |-> (/) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   =/= wne 2367   (/)c0 3450   ifcif 3561   U.cuni 3839   |-> cmpt 4094   omcom 4626   ` cfv 5258   1oc1o 6467  ℕ_∞xnninf 7185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-1o 6474

This theorem is referenced by:  exmidsbthrlem  15666