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Theorem peano3nninf 14759
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. 
|->  ( i  e.  om  |->  if ( i  =  (/) ,  1o ,  ( p `
 U. i ) ) ) )
Assertion
Ref Expression
peano3nninf  |-  ( A  e.  ->  ( 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
StepHypRef Expression
1 fveq1 5515 . . . . . . . . . 10  |-  ( p  =  A  ->  (
p `  U. i )  =  ( A `  U. i ) )
21ifeq2d 3553 . . . . . . . . 9  |-  ( p  =  A  ->  if ( i  =  (/) ,  1o ,  ( p `
 U. i ) )  =  if ( i  =  (/) ,  1o ,  ( A `  U. i ) ) )
32mpteq2dv 4095 . . . . . . . 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. 
|->  ( i  e.  om  |->  if ( i  =  (/) ,  1o ,  ( p `
 U. i ) ) ) )
5 omex 4593 . . . . . . . . 9  |-  om  e.  _V
65mptex 5743 . . . . . . . 8  |-  ( i  e.  om  |->  if ( i  =  (/) ,  1o ,  ( A `  U. i ) ) )  e.  _V
73, 4, 6fvmpt 5594 . . . . . . 7  |-  ( A  e.  ->  ( S `  A
)  =  ( i  e.  om  |->  if ( i  =  (/) ,  1o ,  ( A `  U. i ) ) ) )
8 eqeq1 2184 . . . . . . . . 9  |-  ( i  =  (/)  ->  ( i  =  (/)  <->  (/)  =  (/) ) )
9 unieq 3819 . . . . . . . . . 10  |-  ( i  =  (/)  ->  U. i  =  U. (/) )
109fveq2d 5520 . . . . . . . . 9  |-  ( i  =  (/)  ->  ( A `
 U. i )  =  ( A `  U. (/) ) )
118, 10ifbieq2d 3559 . . . . . . . 8  |-  ( i  =  (/)  ->  if ( i  =  (/) ,  1o ,  ( A `  U. i ) )  =  if ( (/)  =  (/) ,  1o ,  ( A `
 U. (/) ) ) )
1211adantl 277 . . . . . . 7  |-  ( ( A  e.  /\  i  =  (/) )  ->  if ( i  =  (/) ,  1o , 
( A `  U. i ) )  =  if ( (/)  =  (/) ,  1o ,  ( A `
 U. (/) ) ) )
13 peano1 4594 . . . . . . . 8  |-  (/)  e.  om
1413a1i 9 . . . . . . 7  |-  ( A  e.  -> 
(/)  e.  om )
15 eqid 2177 . . . . . . . . . 10  |-  (/)  =  (/)
1615iftruei 3541 . . . . . . . . 9  |-  if (
(/)  =  (/) ,  1o ,  ( A `  U. (/) ) )  =  1o
17 1onn 6521 . . . . . . . . 9  |-  1o  e.  om
1816, 17eqeltri 2250 . . . . . . . 8  |-  if (
(/)  =  (/) ,  1o ,  ( A `  U. (/) ) )  e. 
om
1918a1i 9 . . . . . . 7  |-  ( A  e.  ->  if ( (/)  =  (/) ,  1o ,  ( A `
 U. (/) ) )  e.  om )
207, 12, 14, 19fvmptd 5598 . . . . . 6  |-  ( A  e.  ->  ( ( S `  A ) `  (/) )  =  if ( (/)  =  (/) ,  1o ,  ( A `
 U. (/) ) ) )
2120, 16eqtrdi 2226 . . . . 5  |-  ( A  e.  ->  ( ( S `  A ) `  (/) )  =  1o )
2221adantr 276 . . . 4  |-  ( ( A  e.  /\  ( S `  A )  =  ( x  e.  om  |->  (/) ) )  ->  (
( S `  A
) `  (/) )  =  1o )
23 fveq1 5515 . . . . . 6  |-  ( ( S `  A )  =  ( x  e. 
om  |->  (/) )  ->  (
( S `  A
) `  (/) )  =  ( ( x  e. 
om  |->  (/) ) `  (/) ) )
2423adantl 277 . . . . 5  |-  ( ( A  e.  /\  ( S `  A )  =  ( x  e.  om  |->  (/) ) )  ->  (
( S `  A
) `  (/) )  =  ( ( x  e. 
om  |->  (/) ) `  (/) ) )
2515a1i 9 . . . . . . 7  |-  ( x  =  (/)  ->  (/)  =  (/) )
26 eqid 2177 . . . . . . 7  |-  ( x  e.  om  |->  (/) )  =  ( x  e.  om  |->  (/) )
2725, 26fvmptg 5593 . . . . . 6  |-  ( (
(/)  e.  om  /\  (/)  e.  om )  ->  ( ( x  e.  om  |->  (/) ) `  (/) )  =  (/) )
2813, 13, 27mp2an 426 . . . . 5  |-  ( ( x  e.  om  |->  (/) ) `  (/) )  =  (/)
2924, 28eqtrdi 2226 . . . 4  |-  ( ( A  e.  /\  ( S `  A )  =  ( x  e.  om  |->  (/) ) )  ->  (
( S `  A
) `  (/) )  =  (/) )
3022, 29eqtr3d 2212 . . 3  |-  ( ( A  e.  /\  ( S `  A )  =  ( x  e.  om  |->  (/) ) )  ->  1o  =  (/) )
31 1n0 6433 . . . . 5  |-  1o  =/=  (/)
3231neii 2349 . . . 4  |-  -.  1o  =  (/)
3332a1i 9 . . 3  |-  ( ( A  e.  /\  ( S `  A )  =  ( x  e.  om  |->  (/) ) )  ->  -.  1o  =  (/) )
3430, 33pm2.65da 661 . 2  |-  ( A  e.  ->  -.  ( S `  A )  =  ( x  e.  om  |->  (/) ) )
3534neqned 2354 1  |-  ( A  e.  ->  ( S `  A
)  =/=  ( x  e.  om  |->  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148    =/= wne 2347   (/)c0 3423   ifcif 3535   U.cuni 3810    |-> cmpt 4065   omcom 4590   ` cfv 5217   1oc1o 6410  ℕxnninf 7118
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-1o 6417
This theorem is referenced by:  exmidsbthrlem  14773
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