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Theorem nneneq 7249
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 4175 . . . . . 6  |-  ( x  =  (/)  ->  ( x 
~~  z  <->  (/)  ~~  z
) )
2 eqeq1 2410 . . . . . 6  |-  ( x  =  (/)  ->  ( x  =  z  <->  (/)  =  z ) )
31, 2imbi12d 312 . . . . 5  |-  ( x  =  (/)  ->  ( ( x  ~~  z  ->  x  =  z )  <->  (
(/)  ~~  z  ->  (/)  =  z ) ) )
43ralbidv 2686 . . . 4  |-  ( x  =  (/)  ->  ( A. z  e.  om  (
x  ~~  z  ->  x  =  z )  <->  A. z  e.  om  ( (/)  ~~  z  -> 
(/)  =  z ) ) )
5 breq1 4175 . . . . . 6  |-  ( x  =  y  ->  (
x  ~~  z  <->  y  ~~  z ) )
6 eqeq1 2410 . . . . . 6  |-  ( x  =  y  ->  (
x  =  z  <->  y  =  z ) )
75, 6imbi12d 312 . . . . 5  |-  ( x  =  y  ->  (
( x  ~~  z  ->  x  =  z )  <-> 
( y  ~~  z  ->  y  =  z ) ) )
87ralbidv 2686 . . . 4  |-  ( x  =  y  ->  ( A. z  e.  om  ( x  ~~  z  ->  x  =  z )  <->  A. z  e.  om  (
y  ~~  z  ->  y  =  z ) ) )
9 breq1 4175 . . . . . 6  |-  ( x  =  suc  y  -> 
( x  ~~  z  <->  suc  y  ~~  z ) )
10 eqeq1 2410 . . . . . 6  |-  ( x  =  suc  y  -> 
( x  =  z  <->  suc  y  =  z
) )
119, 10imbi12d 312 . . . . 5  |-  ( x  =  suc  y  -> 
( ( x  ~~  z  ->  x  =  z )  <->  ( suc  y  ~~  z  ->  suc  y  =  z ) ) )
1211ralbidv 2686 . . . 4  |-  ( x  =  suc  y  -> 
( A. z  e. 
om  ( x  ~~  z  ->  x  =  z )  <->  A. z  e.  om  ( suc  y  ~~  z  ->  suc  y  =  z ) ) )
13 breq1 4175 . . . . . 6  |-  ( x  =  A  ->  (
x  ~~  z  <->  A  ~~  z ) )
14 eqeq1 2410 . . . . . 6  |-  ( x  =  A  ->  (
x  =  z  <->  A  =  z ) )
1513, 14imbi12d 312 . . . . 5  |-  ( x  =  A  ->  (
( x  ~~  z  ->  x  =  z )  <-> 
( A  ~~  z  ->  A  =  z ) ) )
1615ralbidv 2686 . . . 4  |-  ( x  =  A  ->  ( A. z  e.  om  ( x  ~~  z  ->  x  =  z )  <->  A. z  e.  om  ( A  ~~  z  ->  A  =  z ) ) )
17 ensym 7115 . . . . . 6  |-  ( (/)  ~~  z  ->  z  ~~  (/) )
18 en0 7129 . . . . . . 7  |-  ( z 
~~  (/)  <->  z  =  (/) )
19 eqcom 2406 . . . . . . 7  |-  ( z  =  (/)  <->  (/)  =  z )
2018, 19bitri 241 . . . . . 6  |-  ( z 
~~  (/)  <->  (/)  =  z )
2117, 20sylib 189 . . . . 5  |-  ( (/)  ~~  z  ->  (/)  =  z )
2221rgenw 2733 . . . 4  |-  A. z  e.  om  ( (/)  ~~  z  -> 
(/)  =  z )
23 nn0suc 4828 . . . . . . 7  |-  ( w  e.  om  ->  (
w  =  (/)  \/  E. z  e.  om  w  =  suc  z ) )
24 en0 7129 . . . . . . . . . . . 12  |-  ( suc  y  ~~  (/)  <->  suc  y  =  (/) )
25 breq2 4176 . . . . . . . . . . . . 13  |-  ( w  =  (/)  ->  ( suc  y  ~~  w  <->  suc  y  ~~  (/) ) )
26 eqeq2 2413 . . . . . . . . . . . . 13  |-  ( w  =  (/)  ->  ( suc  y  =  w  <->  suc  y  =  (/) ) )
2725, 26bibi12d 313 . . . . . . . . . . . 12  |-  ( w  =  (/)  ->  ( ( suc  y  ~~  w  <->  suc  y  =  w )  <-> 
( suc  y  ~~  (/)  <->  suc  y  =  (/) ) ) )
2824, 27mpbiri 225 . . . . . . . . . . 11  |-  ( w  =  (/)  ->  ( suc  y  ~~  w  <->  suc  y  =  w ) )
2928biimpd 199 . . . . . . . . . 10  |-  ( w  =  (/)  ->  ( suc  y  ~~  w  ->  suc  y  =  w
) )
3029a1i 11 . . . . . . . . 9  |-  ( ( y  e.  om  /\  A. z  e.  om  (
y  ~~  z  ->  y  =  z ) )  ->  ( w  =  (/)  ->  ( suc  y  ~~  w  ->  suc  y  =  w ) ) )
31 nfv 1626 . . . . . . . . . . 11  |-  F/ z  y  e.  om
32 nfra1 2716 . . . . . . . . . . 11  |-  F/ z A. z  e.  om  ( y  ~~  z  ->  y  =  z )
3331, 32nfan 1842 . . . . . . . . . 10  |-  F/ z ( y  e.  om  /\ 
A. z  e.  om  ( y  ~~  z  ->  y  =  z ) )
34 nfv 1626 . . . . . . . . . 10  |-  F/ z ( suc  y  ~~  w  ->  suc  y  =  w )
35 rsp 2726 . . . . . . . . . . . . . 14  |-  ( A. z  e.  om  (
y  ~~  z  ->  y  =  z )  -> 
( z  e.  om  ->  ( y  ~~  z  ->  y  =  z ) ) )
36 vex 2919 . . . . . . . . . . . . . . . . . 18  |-  y  e. 
_V
37 vex 2919 . . . . . . . . . . . . . . . . . 18  |-  z  e. 
_V
3836, 37phplem4 7248 . . . . . . . . . . . . . . . . 17  |-  ( ( y  e.  om  /\  z  e.  om )  ->  ( suc  y  ~~  suc  z  ->  y  ~~  z ) )
3938imim1d 71 . . . . . . . . . . . . . . . 16  |-  ( ( y  e.  om  /\  z  e.  om )  ->  ( ( y  ~~  z  ->  y  =  z )  ->  ( suc  y  ~~  suc  z  -> 
y  =  z ) ) )
4039ex 424 . . . . . . . . . . . . . . 15  |-  ( y  e.  om  ->  (
z  e.  om  ->  ( ( y  ~~  z  ->  y  =  z )  ->  ( suc  y  ~~  suc  z  ->  y  =  z ) ) ) )
4140a2d 24 . . . . . . . . . . . . . 14  |-  ( y  e.  om  ->  (
( z  e.  om  ->  ( y  ~~  z  ->  y  =  z ) )  ->  ( z  e.  om  ->  ( suc  y  ~~  suc  z  -> 
y  =  z ) ) ) )
4235, 41syl5 30 . . . . . . . . . . . . 13  |-  ( y  e.  om  ->  ( A. z  e.  om  ( y  ~~  z  ->  y  =  z )  ->  ( z  e. 
om  ->  ( suc  y  ~~  suc  z  ->  y  =  z ) ) ) )
4342imp 419 . . . . . . . . . . . 12  |-  ( ( y  e.  om  /\  A. z  e.  om  (
y  ~~  z  ->  y  =  z ) )  ->  ( z  e. 
om  ->  ( suc  y  ~~  suc  z  ->  y  =  z ) ) )
44 suceq 4606 . . . . . . . . . . . 12  |-  ( y  =  z  ->  suc  y  =  suc  z )
4543, 44syl8 67 . . . . . . . . . . 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 4176 . . . . . . . . . . . . 13  |-  ( w  =  suc  z  -> 
( suc  y  ~~  w 
<->  suc  y  ~~  suc  z ) )
47 eqeq2 2413 . . . . . . . . . . . . 13  |-  ( w  =  suc  z  -> 
( suc  y  =  w 
<->  suc  y  =  suc  z ) )
4846, 47imbi12d 312 . . . . . . . . . . . 12  |-  ( w  =  suc  z  -> 
( ( suc  y  ~~  w  ->  suc  y  =  w )  <->  ( suc  y  ~~  suc  z  ->  suc  y  =  suc  z ) ) )
4948biimprcd 217 . . . . . . . . . . 11  |-  ( ( suc  y  ~~  suc  z  ->  suc  y  =  suc  z )  ->  (
w  =  suc  z  ->  ( suc  y  ~~  w  ->  suc  y  =  w ) ) )
5045, 49syl6 31 . . . . . . . . . 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 2787 . . . . . . . . 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 370 . . . . . . . 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 424 . . . . . . 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 65 . . . . . 6  |-  ( y  e.  om  ->  ( A. z  e.  om  ( y  ~~  z  ->  y  =  z )  ->  ( w  e. 
om  ->  ( suc  y  ~~  w  ->  suc  y  =  w ) ) ) )
5554ralrimdv 2755 . . . . 5  |-  ( y  e.  om  ->  ( A. z  e.  om  ( y  ~~  z  ->  y  =  z )  ->  A. w  e.  om  ( suc  y  ~~  w  ->  suc  y  =  w ) ) )
56 breq2 4176 . . . . . . 7  |-  ( w  =  z  ->  ( suc  y  ~~  w  <->  suc  y  ~~  z ) )
57 eqeq2 2413 . . . . . . 7  |-  ( w  =  z  ->  ( suc  y  =  w  <->  suc  y  =  z ) )
5856, 57imbi12d 312 . . . . . 6  |-  ( w  =  z  ->  (
( suc  y  ~~  w  ->  suc  y  =  w )  <->  ( suc  y  ~~  z  ->  suc  y  =  z )
) )
5958cbvralv 2892 . . . . 5  |-  ( A. w  e.  om  ( suc  y  ~~  w  ->  suc  y  =  w
)  <->  A. z  e.  om  ( suc  y  ~~  z  ->  suc  y  =  z ) )
6055, 59syl6ib 218 . . . 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 4830 . . 3  |-  ( A  e.  om  ->  A. z  e.  om  ( A  ~~  z  ->  A  =  z ) )
62 breq2 4176 . . . . 5  |-  ( z  =  B  ->  ( A  ~~  z  <->  A  ~~  B ) )
63 eqeq2 2413 . . . . 5  |-  ( z  =  B  ->  ( A  =  z  <->  A  =  B ) )
6462, 63imbi12d 312 . . . 4  |-  ( z  =  B  ->  (
( A  ~~  z  ->  A  =  z )  <-> 
( A  ~~  B  ->  A  =  B ) ) )
6564rspcv 3008 . . 3  |-  ( B  e.  om  ->  ( A. z  e.  om  ( A  ~~  z  ->  A  =  z )  ->  ( A  ~~  B  ->  A  =  B ) ) )
6661, 65mpan9 456 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  ~~  B  ->  A  =  B ) )
67 eqeng 7100 . . 3  |-  ( A  e.  om  ->  ( A  =  B  ->  A 
~~  B ) )
6867adantr 452 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  =  B  ->  A  ~~  B
) )
6966, 68impbid 184 1  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  ~~  B  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   (/)c0 3588   class class class wbr 4172   suc csuc 4543   omcom 4804    ~~ cen 7065
This theorem is referenced by:  php  7250  onomeneq  7255  nnsdomo  7260  fineqvlem  7282  dif1enOLD  7299  dif1en  7300  findcard2  7307  cardnn  7806
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-er 6864  df-en 7069
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