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Theorem nneneq 7044
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 4026 . . . . . 6  |-  ( x  =  (/)  ->  ( x 
~~  z  <->  (/)  ~~  z
) )
2 eqeq1 2289 . . . . . 6  |-  ( x  =  (/)  ->  ( x  =  z  <->  (/)  =  z ) )
31, 2imbi12d 311 . . . . 5  |-  ( x  =  (/)  ->  ( ( x  ~~  z  ->  x  =  z )  <->  (
(/)  ~~  z  ->  (/)  =  z ) ) )
43ralbidv 2563 . . . 4  |-  ( x  =  (/)  ->  ( A. z  e.  om  (
x  ~~  z  ->  x  =  z )  <->  A. z  e.  om  ( (/)  ~~  z  -> 
(/)  =  z ) ) )
5 breq1 4026 . . . . . 6  |-  ( x  =  y  ->  (
x  ~~  z  <->  y  ~~  z ) )
6 eqeq1 2289 . . . . . 6  |-  ( x  =  y  ->  (
x  =  z  <->  y  =  z ) )
75, 6imbi12d 311 . . . . 5  |-  ( x  =  y  ->  (
( x  ~~  z  ->  x  =  z )  <-> 
( y  ~~  z  ->  y  =  z ) ) )
87ralbidv 2563 . . . 4  |-  ( x  =  y  ->  ( A. z  e.  om  ( x  ~~  z  ->  x  =  z )  <->  A. z  e.  om  (
y  ~~  z  ->  y  =  z ) ) )
9 breq1 4026 . . . . . 6  |-  ( x  =  suc  y  -> 
( x  ~~  z  <->  suc  y  ~~  z ) )
10 eqeq1 2289 . . . . . 6  |-  ( x  =  suc  y  -> 
( x  =  z  <->  suc  y  =  z
) )
119, 10imbi12d 311 . . . . 5  |-  ( x  =  suc  y  -> 
( ( x  ~~  z  ->  x  =  z )  <->  ( suc  y  ~~  z  ->  suc  y  =  z ) ) )
1211ralbidv 2563 . . . 4  |-  ( x  =  suc  y  -> 
( A. z  e. 
om  ( x  ~~  z  ->  x  =  z )  <->  A. z  e.  om  ( suc  y  ~~  z  ->  suc  y  =  z ) ) )
13 breq1 4026 . . . . . 6  |-  ( x  =  A  ->  (
x  ~~  z  <->  A  ~~  z ) )
14 eqeq1 2289 . . . . . 6  |-  ( x  =  A  ->  (
x  =  z  <->  A  =  z ) )
1513, 14imbi12d 311 . . . . 5  |-  ( x  =  A  ->  (
( x  ~~  z  ->  x  =  z )  <-> 
( A  ~~  z  ->  A  =  z ) ) )
1615ralbidv 2563 . . . 4  |-  ( x  =  A  ->  ( A. z  e.  om  ( x  ~~  z  ->  x  =  z )  <->  A. z  e.  om  ( A  ~~  z  ->  A  =  z ) ) )
17 ensym 6910 . . . . . 6  |-  ( (/)  ~~  z  ->  z  ~~  (/) )
18 en0 6924 . . . . . . 7  |-  ( z 
~~  (/)  <->  z  =  (/) )
19 eqcom 2285 . . . . . . 7  |-  ( z  =  (/)  <->  (/)  =  z )
2018, 19bitri 240 . . . . . 6  |-  ( z 
~~  (/)  <->  (/)  =  z )
2117, 20sylib 188 . . . . 5  |-  ( (/)  ~~  z  ->  (/)  =  z )
2221rgenw 2610 . . . 4  |-  A. z  e.  om  ( (/)  ~~  z  -> 
(/)  =  z )
23 nn0suc 4680 . . . . . . 7  |-  ( w  e.  om  ->  (
w  =  (/)  \/  E. z  e.  om  w  =  suc  z ) )
24 en0 6924 . . . . . . . . . . . 12  |-  ( suc  y  ~~  (/)  <->  suc  y  =  (/) )
25 breq2 4027 . . . . . . . . . . . . 13  |-  ( w  =  (/)  ->  ( suc  y  ~~  w  <->  suc  y  ~~  (/) ) )
26 eqeq2 2292 . . . . . . . . . . . . 13  |-  ( w  =  (/)  ->  ( suc  y  =  w  <->  suc  y  =  (/) ) )
2725, 26bibi12d 312 . . . . . . . . . . . 12  |-  ( w  =  (/)  ->  ( ( suc  y  ~~  w  <->  suc  y  =  w )  <-> 
( suc  y  ~~  (/)  <->  suc  y  =  (/) ) ) )
2824, 27mpbiri 224 . . . . . . . . . . 11  |-  ( w  =  (/)  ->  ( suc  y  ~~  w  <->  suc  y  =  w ) )
2928biimpd 198 . . . . . . . . . 10  |-  ( w  =  (/)  ->  ( suc  y  ~~  w  ->  suc  y  =  w
) )
3029a1i 10 . . . . . . . . 9  |-  ( ( y  e.  om  /\  A. z  e.  om  (
y  ~~  z  ->  y  =  z ) )  ->  ( w  =  (/)  ->  ( suc  y  ~~  w  ->  suc  y  =  w ) ) )
31 nfv 1605 . . . . . . . . . . 11  |-  F/ z  y  e.  om
32 nfra1 2593 . . . . . . . . . . 11  |-  F/ z A. z  e.  om  ( y  ~~  z  ->  y  =  z )
3331, 32nfan 1771 . . . . . . . . . 10  |-  F/ z ( y  e.  om  /\ 
A. z  e.  om  ( y  ~~  z  ->  y  =  z ) )
34 nfv 1605 . . . . . . . . . 10  |-  F/ z ( suc  y  ~~  w  ->  suc  y  =  w )
35 rsp 2603 . . . . . . . . . . . . . 14  |-  ( A. z  e.  om  (
y  ~~  z  ->  y  =  z )  -> 
( z  e.  om  ->  ( y  ~~  z  ->  y  =  z ) ) )
36 vex 2791 . . . . . . . . . . . . . . . . . 18  |-  y  e. 
_V
37 vex 2791 . . . . . . . . . . . . . . . . . 18  |-  z  e. 
_V
3836, 37phplem4 7043 . . . . . . . . . . . . . . . . 17  |-  ( ( y  e.  om  /\  z  e.  om )  ->  ( suc  y  ~~  suc  z  ->  y  ~~  z ) )
3938imim1d 69 . . . . . . . . . . . . . . . 16  |-  ( ( y  e.  om  /\  z  e.  om )  ->  ( ( y  ~~  z  ->  y  =  z )  ->  ( suc  y  ~~  suc  z  -> 
y  =  z ) ) )
4039ex 423 . . . . . . . . . . . . . . 15  |-  ( y  e.  om  ->  (
z  e.  om  ->  ( ( y  ~~  z  ->  y  =  z )  ->  ( suc  y  ~~  suc  z  ->  y  =  z ) ) ) )
4140a2d 23 . . . . . . . . . . . . . 14  |-  ( y  e.  om  ->  (
( z  e.  om  ->  ( y  ~~  z  ->  y  =  z ) )  ->  ( z  e.  om  ->  ( suc  y  ~~  suc  z  -> 
y  =  z ) ) ) )
4235, 41syl5 28 . . . . . . . . . . . . 13  |-  ( y  e.  om  ->  ( A. z  e.  om  ( y  ~~  z  ->  y  =  z )  ->  ( z  e. 
om  ->  ( suc  y  ~~  suc  z  ->  y  =  z ) ) ) )
4342imp 418 . . . . . . . . . . . 12  |-  ( ( y  e.  om  /\  A. z  e.  om  (
y  ~~  z  ->  y  =  z ) )  ->  ( z  e. 
om  ->  ( suc  y  ~~  suc  z  ->  y  =  z ) ) )
44 suceq 4457 . . . . . . . . . . . 12  |-  ( y  =  z  ->  suc  y  =  suc  z )
4543, 44syl8 65 . . . . . . . . . . 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 4027 . . . . . . . . . . . . 13  |-  ( w  =  suc  z  -> 
( suc  y  ~~  w 
<->  suc  y  ~~  suc  z ) )
47 eqeq2 2292 . . . . . . . . . . . . 13  |-  ( w  =  suc  z  -> 
( suc  y  =  w 
<->  suc  y  =  suc  z ) )
4846, 47imbi12d 311 . . . . . . . . . . . 12  |-  ( w  =  suc  z  -> 
( ( suc  y  ~~  w  ->  suc  y  =  w )  <->  ( suc  y  ~~  suc  z  ->  suc  y  =  suc  z ) ) )
4948biimprcd 216 . . . . . . . . . . 11  |-  ( ( suc  y  ~~  suc  z  ->  suc  y  =  suc  z )  ->  (
w  =  suc  z  ->  ( suc  y  ~~  w  ->  suc  y  =  w ) ) )
5045, 49syl6 29 . . . . . . . . . 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 2664 . . . . . . . . 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 369 . . . . . . . 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 423 . . . . . . 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 63 . . . . . 6  |-  ( y  e.  om  ->  ( A. z  e.  om  ( y  ~~  z  ->  y  =  z )  ->  ( w  e. 
om  ->  ( suc  y  ~~  w  ->  suc  y  =  w ) ) ) )
5554ralrimdv 2632 . . . . 5  |-  ( y  e.  om  ->  ( A. z  e.  om  ( y  ~~  z  ->  y  =  z )  ->  A. w  e.  om  ( suc  y  ~~  w  ->  suc  y  =  w ) ) )
56 breq2 4027 . . . . . . 7  |-  ( w  =  z  ->  ( suc  y  ~~  w  <->  suc  y  ~~  z ) )
57 eqeq2 2292 . . . . . . 7  |-  ( w  =  z  ->  ( suc  y  =  w  <->  suc  y  =  z ) )
5856, 57imbi12d 311 . . . . . 6  |-  ( w  =  z  ->  (
( suc  y  ~~  w  ->  suc  y  =  w )  <->  ( suc  y  ~~  z  ->  suc  y  =  z )
) )
5958cbvralv 2764 . . . . 5  |-  ( A. w  e.  om  ( suc  y  ~~  w  ->  suc  y  =  w
)  <->  A. z  e.  om  ( suc  y  ~~  z  ->  suc  y  =  z ) )
6055, 59syl6ib 217 . . . 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 4682 . . 3  |-  ( A  e.  om  ->  A. z  e.  om  ( A  ~~  z  ->  A  =  z ) )
62 breq2 4027 . . . . 5  |-  ( z  =  B  ->  ( A  ~~  z  <->  A  ~~  B ) )
63 eqeq2 2292 . . . . 5  |-  ( z  =  B  ->  ( A  =  z  <->  A  =  B ) )
6462, 63imbi12d 311 . . . 4  |-  ( z  =  B  ->  (
( A  ~~  z  ->  A  =  z )  <-> 
( A  ~~  B  ->  A  =  B ) ) )
6564rspcv 2880 . . 3  |-  ( B  e.  om  ->  ( A. z  e.  om  ( A  ~~  z  ->  A  =  z )  ->  ( A  ~~  B  ->  A  =  B ) ) )
6661, 65mpan9 455 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  ~~  B  ->  A  =  B ) )
67 eqeng 6895 . . 3  |-  ( A  e.  om  ->  ( A  =  B  ->  A 
~~  B ) )
6867adantr 451 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  =  B  ->  A  ~~  B
) )
6966, 68impbid 183 1  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  ~~  B  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   (/)c0 3455   class class class wbr 4023   suc csuc 4394   omcom 4656    ~~ cen 6860
This theorem is referenced by:  php  7045  onomeneq  7050  nnsdomo  7055  fineqvlem  7077  dif1enOLD  7090  dif1en  7091  findcard2  7098  cardnn  7596
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-er 6660  df-en 6864
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