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Theorem nneneq 6929
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
StepHypRef Expression
1 breq1 3923 . . . . . 6  |-  ( x  =  (/)  ->  ( x 
~~  z  <->  (/)  ~~  z
) )
2 eqeq1 2259 . . . . . 6  |-  ( x  =  (/)  ->  ( x  =  z  <->  (/)  =  z ) )
31, 2imbi12d 313 . . . . 5  |-  ( x  =  (/)  ->  ( ( x  ~~  z  ->  x  =  z )  <->  (
(/)  ~~  z  ->  (/)  =  z ) ) )
43ralbidv 2527 . . . 4  |-  ( x  =  (/)  ->  ( A. z  e.  om  (
x  ~~  z  ->  x  =  z )  <->  A. z  e.  om  ( (/)  ~~  z  -> 
(/)  =  z ) ) )
5 breq1 3923 . . . . . 6  |-  ( x  =  y  ->  (
x  ~~  z  <->  y  ~~  z ) )
6 eqeq1 2259 . . . . . 6  |-  ( x  =  y  ->  (
x  =  z  <->  y  =  z ) )
75, 6imbi12d 313 . . . . 5  |-  ( x  =  y  ->  (
( x  ~~  z  ->  x  =  z )  <-> 
( y  ~~  z  ->  y  =  z ) ) )
87ralbidv 2527 . . . 4  |-  ( x  =  y  ->  ( A. z  e.  om  ( x  ~~  z  ->  x  =  z )  <->  A. z  e.  om  (
y  ~~  z  ->  y  =  z ) ) )
9 breq1 3923 . . . . . 6  |-  ( x  =  suc  y  -> 
( x  ~~  z  <->  suc  y  ~~  z ) )
10 eqeq1 2259 . . . . . 6  |-  ( x  =  suc  y  -> 
( x  =  z  <->  suc  y  =  z
) )
119, 10imbi12d 313 . . . . 5  |-  ( x  =  suc  y  -> 
( ( x  ~~  z  ->  x  =  z )  <->  ( suc  y  ~~  z  ->  suc  y  =  z ) ) )
1211ralbidv 2527 . . . 4  |-  ( x  =  suc  y  -> 
( A. z  e. 
om  ( x  ~~  z  ->  x  =  z )  <->  A. z  e.  om  ( suc  y  ~~  z  ->  suc  y  =  z ) ) )
13 breq1 3923 . . . . . 6  |-  ( x  =  A  ->  (
x  ~~  z  <->  A  ~~  z ) )
14 eqeq1 2259 . . . . . 6  |-  ( x  =  A  ->  (
x  =  z  <->  A  =  z ) )
1513, 14imbi12d 313 . . . . 5  |-  ( x  =  A  ->  (
( x  ~~  z  ->  x  =  z )  <-> 
( A  ~~  z  ->  A  =  z ) ) )
1615ralbidv 2527 . . . 4  |-  ( x  =  A  ->  ( A. z  e.  om  ( x  ~~  z  ->  x  =  z )  <->  A. z  e.  om  ( A  ~~  z  ->  A  =  z ) ) )
17 ensym 6796 . . . . . 6  |-  ( (/)  ~~  z  ->  z  ~~  (/) )
18 en0 6809 . . . . . . 7  |-  ( z 
~~  (/)  <->  z  =  (/) )
19 eqcom 2255 . . . . . . 7  |-  ( z  =  (/)  <->  (/)  =  z )
2018, 19bitri 242 . . . . . 6  |-  ( z 
~~  (/)  <->  (/)  =  z )
2117, 20sylib 190 . . . . 5  |-  ( (/)  ~~  z  ->  (/)  =  z )
2221rgenw 2572 . . . 4  |-  A. z  e.  om  ( (/)  ~~  z  -> 
(/)  =  z )
23 nn0suc 4571 . . . . . . 7  |-  ( w  e.  om  ->  (
w  =  (/)  \/  E. z  e.  om  w  =  suc  z ) )
24 en0 6809 . . . . . . . . . . . 12  |-  ( suc  y  ~~  (/)  <->  suc  y  =  (/) )
25 breq2 3924 . . . . . . . . . . . . 13  |-  ( w  =  (/)  ->  ( suc  y  ~~  w  <->  suc  y  ~~  (/) ) )
26 eqeq2 2262 . . . . . . . . . . . . 13  |-  ( w  =  (/)  ->  ( suc  y  =  w  <->  suc  y  =  (/) ) )
2725, 26bibi12d 314 . . . . . . . . . . . 12  |-  ( w  =  (/)  ->  ( ( suc  y  ~~  w  <->  suc  y  =  w )  <-> 
( suc  y  ~~  (/)  <->  suc  y  =  (/) ) ) )
2824, 27mpbiri 226 . . . . . . . . . . 11  |-  ( w  =  (/)  ->  ( suc  y  ~~  w  <->  suc  y  =  w ) )
2928biimpd 200 . . . . . . . . . 10  |-  ( w  =  (/)  ->  ( suc  y  ~~  w  ->  suc  y  =  w
) )
3029a1i 12 . . . . . . . . 9  |-  ( ( y  e.  om  /\  A. z  e.  om  (
y  ~~  z  ->  y  =  z ) )  ->  ( w  =  (/)  ->  ( suc  y  ~~  w  ->  suc  y  =  w ) ) )
31 nfv 1629 . . . . . . . . . . 11  |-  F/ z  y  e.  om
32 nfra1 2555 . . . . . . . . . . 11  |-  F/ z A. z  e.  om  ( y  ~~  z  ->  y  =  z )
3331, 32nfan 1737 . . . . . . . . . 10  |-  F/ z ( y  e.  om  /\ 
A. z  e.  om  ( y  ~~  z  ->  y  =  z ) )
34 nfv 1629 . . . . . . . . . 10  |-  F/ z ( suc  y  ~~  w  ->  suc  y  =  w )
35 ra4 2565 . . . . . . . . . . . . . 14  |-  ( A. z  e.  om  (
y  ~~  z  ->  y  =  z )  -> 
( z  e.  om  ->  ( y  ~~  z  ->  y  =  z ) ) )
36 vex 2730 . . . . . . . . . . . . . . . . . 18  |-  y  e. 
_V
37 vex 2730 . . . . . . . . . . . . . . . . . 18  |-  z  e. 
_V
3836, 37phplem4 6928 . . . . . . . . . . . . . . . . 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 425 . . . . . . . . . . . . . . 15  |-  ( y  e.  om  ->  (
z  e.  om  ->  ( ( y  ~~  z  ->  y  =  z )  ->  ( suc  y  ~~  suc  z  ->  y  =  z ) ) ) )
4140a2d 25 . . . . . . . . . . . . . 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 420 . . . . . . . . . . . 12  |-  ( ( y  e.  om  /\  A. z  e.  om  (
y  ~~  z  ->  y  =  z ) )  ->  ( z  e. 
om  ->  ( suc  y  ~~  suc  z  ->  y  =  z ) ) )
44 suceq 4350 . . . . . . . . . . . 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 3924 . . . . . . . . . . . . 13  |-  ( w  =  suc  z  -> 
( suc  y  ~~  w 
<->  suc  y  ~~  suc  z ) )
47 eqeq2 2262 . . . . . . . . . . . . 13  |-  ( w  =  suc  z  -> 
( suc  y  =  w 
<->  suc  y  =  suc  z ) )
4846, 47imbi12d 313 . . . . . . . . . . . 12  |-  ( w  =  suc  z  -> 
( ( suc  y  ~~  w  ->  suc  y  =  w )  <->  ( suc  y  ~~  suc  z  ->  suc  y  =  suc  z ) ) )
4948biimprcd 218 . . . . . . . . . . 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 2626 . . . . . . . . 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 371 . . . . . . . 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 425 . . . . . . 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 2594 . . . . 5  |-  ( y  e.  om  ->  ( A. z  e.  om  ( y  ~~  z  ->  y  =  z )  ->  A. w  e.  om  ( suc  y  ~~  w  ->  suc  y  =  w ) ) )
56 breq2 3924 . . . . . . 7  |-  ( w  =  z  ->  ( suc  y  ~~  w  <->  suc  y  ~~  z ) )
57 eqeq2 2262 . . . . . . 7  |-  ( w  =  z  ->  ( suc  y  =  w  <->  suc  y  =  z ) )
5856, 57imbi12d 313 . . . . . 6  |-  ( w  =  z  ->  (
( suc  y  ~~  w  ->  suc  y  =  w )  <->  ( suc  y  ~~  z  ->  suc  y  =  z )
) )
5958cbvralv 2708 . . . . 5  |-  ( A. w  e.  om  ( suc  y  ~~  w  ->  suc  y  =  w
)  <->  A. z  e.  om  ( suc  y  ~~  z  ->  suc  y  =  z ) )
6055, 59syl6ib 219 . . . 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 4573 . . 3  |-  ( A  e.  om  ->  A. z  e.  om  ( A  ~~  z  ->  A  =  z ) )
62 breq2 3924 . . . . 5  |-  ( z  =  B  ->  ( A  ~~  z  <->  A  ~~  B ) )
63 eqeq2 2262 . . . . 5  |-  ( z  =  B  ->  ( A  =  z  <->  A  =  B ) )
6462, 63imbi12d 313 . . . 4  |-  ( z  =  B  ->  (
( A  ~~  z  ->  A  =  z )  <-> 
( A  ~~  B  ->  A  =  B ) ) )
6564rcla4v 2817 . . 3  |-  ( B  e.  om  ->  ( A. z  e.  om  ( A  ~~  z  ->  A  =  z )  ->  ( A  ~~  B  ->  A  =  B ) ) )
6661, 65mpan9 457 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  ~~  B  ->  A  =  B ) )
67 eqeng 6781 . . 3  |-  ( A  e.  om  ->  ( A  =  B  ->  A 
~~  B ) )
6867adantr 453 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  =  B  ->  A  ~~  B
) )
6966, 68impbid 185 1  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  ~~  B  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2509   E.wrex 2510   (/)c0 3362   class class class wbr 3920   suc csuc 4287   omcom 4547    ~~ cen 6746
This theorem is referenced by:  php  6930  onomeneq  6935  nnsdomo  6940  fineqvlem  6962  dif1enOLD  6975  dif1en  6976  findcard2  6983  cardnn  7480
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-er 6546  df-en 6750
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