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Theorem nneneq 6392
Description: Two equinumerous natural numbers are equal. Proposition 10.20 of [TakeutiZaring] p. 90 and its converse. Also compare Corollary 6E of [Enderton] p. 136. (Contributed by NM, 28-May-1998.)
Assertion
Ref Expression
nneneq  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  ~~  B  <->  A  =  B ) )

Proof of Theorem nneneq
Dummy variables  x  y  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 3796 . . . . . 6  |-  ( x  =  (/)  ->  ( x 
~~  z  <->  (/)  ~~  z
) )
2 eqeq1 2088 . . . . . 6  |-  ( x  =  (/)  ->  ( x  =  z  <->  (/)  =  z ) )
31, 2imbi12d 232 . . . . 5  |-  ( x  =  (/)  ->  ( ( x  ~~  z  ->  x  =  z )  <->  (
(/)  ~~  z  ->  (/)  =  z ) ) )
43ralbidv 2369 . . . 4  |-  ( x  =  (/)  ->  ( A. z  e.  om  (
x  ~~  z  ->  x  =  z )  <->  A. z  e.  om  ( (/)  ~~  z  -> 
(/)  =  z ) ) )
5 breq1 3796 . . . . . 6  |-  ( x  =  y  ->  (
x  ~~  z  <->  y  ~~  z ) )
6 eqeq1 2088 . . . . . 6  |-  ( x  =  y  ->  (
x  =  z  <->  y  =  z ) )
75, 6imbi12d 232 . . . . 5  |-  ( x  =  y  ->  (
( x  ~~  z  ->  x  =  z )  <-> 
( y  ~~  z  ->  y  =  z ) ) )
87ralbidv 2369 . . . 4  |-  ( x  =  y  ->  ( A. z  e.  om  ( x  ~~  z  ->  x  =  z )  <->  A. z  e.  om  (
y  ~~  z  ->  y  =  z ) ) )
9 breq1 3796 . . . . . 6  |-  ( x  =  suc  y  -> 
( x  ~~  z  <->  suc  y  ~~  z ) )
10 eqeq1 2088 . . . . . 6  |-  ( x  =  suc  y  -> 
( x  =  z  <->  suc  y  =  z
) )
119, 10imbi12d 232 . . . . 5  |-  ( x  =  suc  y  -> 
( ( x  ~~  z  ->  x  =  z )  <->  ( suc  y  ~~  z  ->  suc  y  =  z ) ) )
1211ralbidv 2369 . . . 4  |-  ( x  =  suc  y  -> 
( A. z  e. 
om  ( x  ~~  z  ->  x  =  z )  <->  A. z  e.  om  ( suc  y  ~~  z  ->  suc  y  =  z ) ) )
13 breq1 3796 . . . . . 6  |-  ( x  =  A  ->  (
x  ~~  z  <->  A  ~~  z ) )
14 eqeq1 2088 . . . . . 6  |-  ( x  =  A  ->  (
x  =  z  <->  A  =  z ) )
1513, 14imbi12d 232 . . . . 5  |-  ( x  =  A  ->  (
( x  ~~  z  ->  x  =  z )  <-> 
( A  ~~  z  ->  A  =  z ) ) )
1615ralbidv 2369 . . . 4  |-  ( x  =  A  ->  ( A. z  e.  om  ( x  ~~  z  ->  x  =  z )  <->  A. z  e.  om  ( A  ~~  z  ->  A  =  z ) ) )
17 ensym 6328 . . . . . 6  |-  ( (/)  ~~  z  ->  z  ~~  (/) )
18 en0 6342 . . . . . . 7  |-  ( z 
~~  (/)  <->  z  =  (/) )
19 eqcom 2084 . . . . . . 7  |-  ( z  =  (/)  <->  (/)  =  z )
2018, 19bitri 182 . . . . . 6  |-  ( z 
~~  (/)  <->  (/)  =  z )
2117, 20sylib 120 . . . . 5  |-  ( (/)  ~~  z  ->  (/)  =  z )
2221rgenw 2419 . . . 4  |-  A. z  e.  om  ( (/)  ~~  z  -> 
(/)  =  z )
23 nn0suc 4353 . . . . . . 7  |-  ( w  e.  om  ->  (
w  =  (/)  \/  E. z  e.  om  w  =  suc  z ) )
24 en0 6342 . . . . . . . . . . . 12  |-  ( suc  y  ~~  (/)  <->  suc  y  =  (/) )
25 breq2 3797 . . . . . . . . . . . . 13  |-  ( w  =  (/)  ->  ( suc  y  ~~  w  <->  suc  y  ~~  (/) ) )
26 eqeq2 2091 . . . . . . . . . . . . 13  |-  ( w  =  (/)  ->  ( suc  y  =  w  <->  suc  y  =  (/) ) )
2725, 26bibi12d 233 . . . . . . . . . . . 12  |-  ( w  =  (/)  ->  ( ( suc  y  ~~  w  <->  suc  y  =  w )  <-> 
( suc  y  ~~  (/)  <->  suc  y  =  (/) ) ) )
2824, 27mpbiri 166 . . . . . . . . . . 11  |-  ( w  =  (/)  ->  ( suc  y  ~~  w  <->  suc  y  =  w ) )
2928biimpd 142 . . . . . . . . . 10  |-  ( w  =  (/)  ->  ( suc  y  ~~  w  ->  suc  y  =  w
) )
3029a1i 9 . . . . . . . . 9  |-  ( ( y  e.  om  /\  A. z  e.  om  (
y  ~~  z  ->  y  =  z ) )  ->  ( w  =  (/)  ->  ( suc  y  ~~  w  ->  suc  y  =  w ) ) )
31 nfv 1462 . . . . . . . . . . 11  |-  F/ z  y  e.  om
32 nfra1 2398 . . . . . . . . . . 11  |-  F/ z A. z  e.  om  ( y  ~~  z  ->  y  =  z )
3331, 32nfan 1498 . . . . . . . . . 10  |-  F/ z ( y  e.  om  /\ 
A. z  e.  om  ( y  ~~  z  ->  y  =  z ) )
34 nfv 1462 . . . . . . . . . 10  |-  F/ z ( suc  y  ~~  w  ->  suc  y  =  w )
35 rsp 2412 . . . . . . . . . . . . . 14  |-  ( A. z  e.  om  (
y  ~~  z  ->  y  =  z )  -> 
( z  e.  om  ->  ( y  ~~  z  ->  y  =  z ) ) )
36 vex 2605 . . . . . . . . . . . . . . . . . 18  |-  y  e. 
_V
37 vex 2605 . . . . . . . . . . . . . . . . . 18  |-  z  e. 
_V
3836, 37phplem4 6390 . . . . . . . . . . . . . . . . 17  |-  ( ( y  e.  om  /\  z  e.  om )  ->  ( suc  y  ~~  suc  z  ->  y  ~~  z ) )
3938imim1d 74 . . . . . . . . . . . . . . . 16  |-  ( ( y  e.  om  /\  z  e.  om )  ->  ( ( y  ~~  z  ->  y  =  z )  ->  ( suc  y  ~~  suc  z  -> 
y  =  z ) ) )
4039ex 113 . . . . . . . . . . . . . . 15  |-  ( y  e.  om  ->  (
z  e.  om  ->  ( ( y  ~~  z  ->  y  =  z )  ->  ( suc  y  ~~  suc  z  ->  y  =  z ) ) ) )
4140a2d 26 . . . . . . . . . . . . . 14  |-  ( y  e.  om  ->  (
( z  e.  om  ->  ( y  ~~  z  ->  y  =  z ) )  ->  ( z  e.  om  ->  ( suc  y  ~~  suc  z  -> 
y  =  z ) ) ) )
4235, 41syl5 32 . . . . . . . . . . . . 13  |-  ( y  e.  om  ->  ( A. z  e.  om  ( y  ~~  z  ->  y  =  z )  ->  ( z  e. 
om  ->  ( suc  y  ~~  suc  z  ->  y  =  z ) ) ) )
4342imp 122 . . . . . . . . . . . 12  |-  ( ( y  e.  om  /\  A. z  e.  om  (
y  ~~  z  ->  y  =  z ) )  ->  ( z  e. 
om  ->  ( suc  y  ~~  suc  z  ->  y  =  z ) ) )
44 suceq 4165 . . . . . . . . . . . 12  |-  ( y  =  z  ->  suc  y  =  suc  z )
4543, 44syl8 70 . . . . . . . . . . 11  |-  ( ( y  e.  om  /\  A. z  e.  om  (
y  ~~  z  ->  y  =  z ) )  ->  ( z  e. 
om  ->  ( suc  y  ~~  suc  z  ->  suc  y  =  suc  z ) ) )
46 breq2 3797 . . . . . . . . . . . . 13  |-  ( w  =  suc  z  -> 
( suc  y  ~~  w 
<->  suc  y  ~~  suc  z ) )
47 eqeq2 2091 . . . . . . . . . . . . 13  |-  ( w  =  suc  z  -> 
( suc  y  =  w 
<->  suc  y  =  suc  z ) )
4846, 47imbi12d 232 . . . . . . . . . . . 12  |-  ( w  =  suc  z  -> 
( ( suc  y  ~~  w  ->  suc  y  =  w )  <->  ( suc  y  ~~  suc  z  ->  suc  y  =  suc  z ) ) )
4948biimprcd 158 . . . . . . . . . . 11  |-  ( ( suc  y  ~~  suc  z  ->  suc  y  =  suc  z )  ->  (
w  =  suc  z  ->  ( suc  y  ~~  w  ->  suc  y  =  w ) ) )
5045, 49syl6 33 . . . . . . . . . 10  |-  ( ( y  e.  om  /\  A. z  e.  om  (
y  ~~  z  ->  y  =  z ) )  ->  ( z  e. 
om  ->  ( w  =  suc  z  ->  ( suc  y  ~~  w  ->  suc  y  =  w
) ) ) )
5133, 34, 50rexlimd 2475 . . . . . . . . 9  |-  ( ( y  e.  om  /\  A. z  e.  om  (
y  ~~  z  ->  y  =  z ) )  ->  ( E. z  e.  om  w  =  suc  z  ->  ( suc  y  ~~  w  ->  suc  y  =  w ) ) )
5230, 51jaod 670 . . . . . . . 8  |-  ( ( y  e.  om  /\  A. z  e.  om  (
y  ~~  z  ->  y  =  z ) )  ->  ( ( w  =  (/)  \/  E. z  e.  om  w  =  suc  z )  ->  ( suc  y  ~~  w  ->  suc  y  =  w
) ) )
5352ex 113 . . . . . . 7  |-  ( y  e.  om  ->  ( A. z  e.  om  ( y  ~~  z  ->  y  =  z )  ->  ( ( w  =  (/)  \/  E. z  e.  om  w  =  suc  z )  ->  ( suc  y  ~~  w  ->  suc  y  =  w
) ) ) )
5423, 53syl7 68 . . . . . 6  |-  ( y  e.  om  ->  ( A. z  e.  om  ( y  ~~  z  ->  y  =  z )  ->  ( w  e. 
om  ->  ( suc  y  ~~  w  ->  suc  y  =  w ) ) ) )
5554ralrimdv 2441 . . . . 5  |-  ( y  e.  om  ->  ( A. z  e.  om  ( y  ~~  z  ->  y  =  z )  ->  A. w  e.  om  ( suc  y  ~~  w  ->  suc  y  =  w ) ) )
56 breq2 3797 . . . . . . 7  |-  ( w  =  z  ->  ( suc  y  ~~  w  <->  suc  y  ~~  z ) )
57 eqeq2 2091 . . . . . . 7  |-  ( w  =  z  ->  ( suc  y  =  w  <->  suc  y  =  z ) )
5856, 57imbi12d 232 . . . . . 6  |-  ( w  =  z  ->  (
( suc  y  ~~  w  ->  suc  y  =  w )  <->  ( suc  y  ~~  z  ->  suc  y  =  z )
) )
5958cbvralv 2578 . . . . 5  |-  ( A. w  e.  om  ( suc  y  ~~  w  ->  suc  y  =  w
)  <->  A. z  e.  om  ( suc  y  ~~  z  ->  suc  y  =  z ) )
6055, 59syl6ib 159 . . . 4  |-  ( y  e.  om  ->  ( A. z  e.  om  ( y  ~~  z  ->  y  =  z )  ->  A. z  e.  om  ( suc  y  ~~  z  ->  suc  y  =  z ) ) )
614, 8, 12, 16, 22, 60finds 4349 . . 3  |-  ( A  e.  om  ->  A. z  e.  om  ( A  ~~  z  ->  A  =  z ) )
62 breq2 3797 . . . . 5  |-  ( z  =  B  ->  ( A  ~~  z  <->  A  ~~  B ) )
63 eqeq2 2091 . . . . 5  |-  ( z  =  B  ->  ( A  =  z  <->  A  =  B ) )
6462, 63imbi12d 232 . . . 4  |-  ( z  =  B  ->  (
( A  ~~  z  ->  A  =  z )  <-> 
( A  ~~  B  ->  A  =  B ) ) )
6564rspcv 2698 . . 3  |-  ( B  e.  om  ->  ( A. z  e.  om  ( A  ~~  z  ->  A  =  z )  ->  ( A  ~~  B  ->  A  =  B ) ) )
6661, 65mpan9 275 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  ~~  B  ->  A  =  B ) )
67 eqeng 6313 . . 3  |-  ( A  e.  om  ->  ( A  =  B  ->  A 
~~  B ) )
6867adantr 270 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  =  B  ->  A  ~~  B
) )
6966, 68impbid 127 1  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  ~~  B  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662    = wceq 1285    e. wcel 1434   A.wral 2349   E.wrex 2350   (/)c0 3258   class class class wbr 3793   suc csuc 4128   omcom 4339    ~~ cen 6285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-opab 3848  df-tr 3884  df-id 4056  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-er 6172  df-en 6288
This theorem is referenced by:  findcard2  6423  findcard2s  6424  unsnfidcex  6440  unsnfidcel  6441  sizeen  9808  sizeunlem  9828
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