Theorem peano3nninf 16332

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 5625	. . . . . . . . . 10 |- ( p = A -> ( p ` U. i ) = ( A ` U. i ) )
2	1	ifeq2d 3621	. . . . . . . . 9 |- ( p = A -> if ( i = (/) , 1o , ( p ` U. i ) ) = if ( i = (/) , 1o , ( A ` U. i ) ) )
3	2	mpteq2dv 4174	. . . . . . . 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 4684	. . . . . . . . 9 |- om e. _V
6	5	mptex 5864	. . . . . . . 8 |- ( i e. om |-> if ( i = (/) , 1o , ( A ` U. i ) ) ) e. _V
7	3, 4, 6	fvmpt 5710	. . . . . . 7 |- ( A e. ℕ∞ -> ( S ` A ) = ( i e. om |-> if ( i = (/) , 1o , ( A ` U. i ) ) ) )
8		eqeq1 2236	. . . . . . . . 9 |- ( i = (/) -> ( i = (/) <-> (/) = (/) ) )
9		unieq 3896	. . . . . . . . . 10 |- ( i = (/) -> U. i = U. (/) )
10	9	fveq2d 5630	. . . . . . . . 9 |- ( i = (/) -> ( A ` U. i ) = ( A ` U. (/) ) )
11	8, 10	ifbieq2d 3627	. . . . . . . 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 4685	. . . . . . . 8 |- (/) e. om
14	13	a1i 9	. . . . . . 7 |- ( A e. ℕ∞ -> (/) e. om )
15		eqid 2229	. . . . . . . . . 10 |- (/) = (/)
16	15	iftruei 3608	. . . . . . . . 9 |- if ( (/) = (/) , 1o , ( A ` U. (/) ) ) = 1o
17		1onn 6664	. . . . . . . . 9 |- 1o e. om
18	16, 17	eqeltri 2302	. . . . . . . 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 5714	. . . . . 6 |- ( A e. ℕ∞ -> ( ( S ` A ) ` (/) ) = if ( (/) = (/) , 1o , ( A ` U. (/) ) ) )
21	20, 16	eqtrdi 2278	. . . . 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 5625	. . . . . 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 2229	. . . . . . 7 |- ( x e. om |-> (/) ) = ( x e. om |-> (/) )
27	25, 26	fvmptg 5709	. . . . . 6 |- ( ( (/) e. om /\ (/) e. om ) -> ( ( x e. om |-> (/) ) ` (/) ) = (/) )
28	13, 13, 27	mp2an 426	. . . . 5 |- ( ( x e. om |-> (/) ) ` (/) ) = (/)
29	24, 28	eqtrdi 2278	. . . 4 |- ( ( A e. ℕ∞ /\ ( S ` A ) = ( x e. om |-> (/) ) ) -> ( ( S ` A ) ` (/) ) = (/) )
30	22, 29	eqtr3d 2264	. . 3 |- ( ( A e. ℕ∞ /\ ( S ` A ) = ( x e. om |-> (/) ) ) -> 1o = (/) )
31		1n0 6576	. . . . 5 |- 1o =/= (/)
32	31	neii 2402	. . . 4 |- -. 1o = (/)
33	32	a1i 9	. . 3 |- ( ( A e. ℕ∞ /\ ( S ` A ) = ( x e. om |-> (/) ) ) -> -. 1o = (/) )
34	30, 33	pm2.65da 665	. 2 |- ( A e. ℕ∞ -> -. ( S ` A ) = ( x e. om |-> (/) ) )
35	34	neqned 2407	1 |- ( A e. ℕ∞ -> ( S ` A ) =/= ( x e. om |-> (/) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   =/= wne 2400   (/)c0 3491   ifcif 3602   U.cuni 3887   |-> cmpt 4144   omcom 4681   ` cfv 5317   1oc1o 6553  ℕ_∞xnninf 7282

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-1o 6560

This theorem is referenced by:  exmidsbthrlem  16349