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Theorem nneneq 6835
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 3992 . . . . . 6  |-  ( x  =  (/)  ->  ( x 
~~  z  <->  (/)  ~~  z
) )
2 eqeq1 2177 . . . . . 6  |-  ( x  =  (/)  ->  ( x  =  z  <->  (/)  =  z ) )
31, 2imbi12d 233 . . . . 5  |-  ( x  =  (/)  ->  ( ( x  ~~  z  ->  x  =  z )  <->  (
(/)  ~~  z  ->  (/)  =  z ) ) )
43ralbidv 2470 . . . 4  |-  ( x  =  (/)  ->  ( A. z  e.  om  (
x  ~~  z  ->  x  =  z )  <->  A. z  e.  om  ( (/)  ~~  z  -> 
(/)  =  z ) ) )
5 breq1 3992 . . . . . 6  |-  ( x  =  y  ->  (
x  ~~  z  <->  y  ~~  z ) )
6 eqeq1 2177 . . . . . 6  |-  ( x  =  y  ->  (
x  =  z  <->  y  =  z ) )
75, 6imbi12d 233 . . . . 5  |-  ( x  =  y  ->  (
( x  ~~  z  ->  x  =  z )  <-> 
( y  ~~  z  ->  y  =  z ) ) )
87ralbidv 2470 . . . 4  |-  ( x  =  y  ->  ( A. z  e.  om  ( x  ~~  z  ->  x  =  z )  <->  A. z  e.  om  (
y  ~~  z  ->  y  =  z ) ) )
9 breq1 3992 . . . . . 6  |-  ( x  =  suc  y  -> 
( x  ~~  z  <->  suc  y  ~~  z ) )
10 eqeq1 2177 . . . . . 6  |-  ( x  =  suc  y  -> 
( x  =  z  <->  suc  y  =  z
) )
119, 10imbi12d 233 . . . . 5  |-  ( x  =  suc  y  -> 
( ( x  ~~  z  ->  x  =  z )  <->  ( suc  y  ~~  z  ->  suc  y  =  z ) ) )
1211ralbidv 2470 . . . 4  |-  ( x  =  suc  y  -> 
( A. z  e. 
om  ( x  ~~  z  ->  x  =  z )  <->  A. z  e.  om  ( suc  y  ~~  z  ->  suc  y  =  z ) ) )
13 breq1 3992 . . . . . 6  |-  ( x  =  A  ->  (
x  ~~  z  <->  A  ~~  z ) )
14 eqeq1 2177 . . . . . 6  |-  ( x  =  A  ->  (
x  =  z  <->  A  =  z ) )
1513, 14imbi12d 233 . . . . 5  |-  ( x  =  A  ->  (
( x  ~~  z  ->  x  =  z )  <-> 
( A  ~~  z  ->  A  =  z ) ) )
1615ralbidv 2470 . . . 4  |-  ( x  =  A  ->  ( A. z  e.  om  ( x  ~~  z  ->  x  =  z )  <->  A. z  e.  om  ( A  ~~  z  ->  A  =  z ) ) )
17 ensym 6759 . . . . . 6  |-  ( (/)  ~~  z  ->  z  ~~  (/) )
18 en0 6773 . . . . . . 7  |-  ( z 
~~  (/)  <->  z  =  (/) )
19 eqcom 2172 . . . . . . 7  |-  ( z  =  (/)  <->  (/)  =  z )
2018, 19bitri 183 . . . . . 6  |-  ( z 
~~  (/)  <->  (/)  =  z )
2117, 20sylib 121 . . . . 5  |-  ( (/)  ~~  z  ->  (/)  =  z )
2221rgenw 2525 . . . 4  |-  A. z  e.  om  ( (/)  ~~  z  -> 
(/)  =  z )
23 nn0suc 4588 . . . . . . 7  |-  ( w  e.  om  ->  (
w  =  (/)  \/  E. z  e.  om  w  =  suc  z ) )
24 en0 6773 . . . . . . . . . . . 12  |-  ( suc  y  ~~  (/)  <->  suc  y  =  (/) )
25 breq2 3993 . . . . . . . . . . . . 13  |-  ( w  =  (/)  ->  ( suc  y  ~~  w  <->  suc  y  ~~  (/) ) )
26 eqeq2 2180 . . . . . . . . . . . . 13  |-  ( w  =  (/)  ->  ( suc  y  =  w  <->  suc  y  =  (/) ) )
2725, 26bibi12d 234 . . . . . . . . . . . 12  |-  ( w  =  (/)  ->  ( ( suc  y  ~~  w  <->  suc  y  =  w )  <-> 
( suc  y  ~~  (/)  <->  suc  y  =  (/) ) ) )
2824, 27mpbiri 167 . . . . . . . . . . 11  |-  ( w  =  (/)  ->  ( suc  y  ~~  w  <->  suc  y  =  w ) )
2928biimpd 143 . . . . . . . . . 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 1521 . . . . . . . . . . 11  |-  F/ z  y  e.  om
32 nfra1 2501 . . . . . . . . . . 11  |-  F/ z A. z  e.  om  ( y  ~~  z  ->  y  =  z )
3331, 32nfan 1558 . . . . . . . . . 10  |-  F/ z ( y  e.  om  /\ 
A. z  e.  om  ( y  ~~  z  ->  y  =  z ) )
34 nfv 1521 . . . . . . . . . 10  |-  F/ z ( suc  y  ~~  w  ->  suc  y  =  w )
35 rsp 2517 . . . . . . . . . . . . . 14  |-  ( A. z  e.  om  (
y  ~~  z  ->  y  =  z )  -> 
( z  e.  om  ->  ( y  ~~  z  ->  y  =  z ) ) )
36 vex 2733 . . . . . . . . . . . . . . . . . 18  |-  y  e. 
_V
37 vex 2733 . . . . . . . . . . . . . . . . . 18  |-  z  e. 
_V
3836, 37phplem4 6833 . . . . . . . . . . . . . . . . 17  |-  ( ( y  e.  om  /\  z  e.  om )  ->  ( suc  y  ~~  suc  z  ->  y  ~~  z ) )
3938imim1d 75 . . . . . . . . . . . . . . . 16  |-  ( ( y  e.  om  /\  z  e.  om )  ->  ( ( y  ~~  z  ->  y  =  z )  ->  ( suc  y  ~~  suc  z  -> 
y  =  z ) ) )
4039ex 114 . . . . . . . . . . . . . . 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 123 . . . . . . . . . . . 12  |-  ( ( y  e.  om  /\  A. z  e.  om  (
y  ~~  z  ->  y  =  z ) )  ->  ( z  e. 
om  ->  ( suc  y  ~~  suc  z  ->  y  =  z ) ) )
44 suceq 4387 . . . . . . . . . . . 12  |-  ( y  =  z  ->  suc  y  =  suc  z )
4543, 44syl8 71 . . . . . . . . . . 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 3993 . . . . . . . . . . . . 13  |-  ( w  =  suc  z  -> 
( suc  y  ~~  w 
<->  suc  y  ~~  suc  z ) )
47 eqeq2 2180 . . . . . . . . . . . . 13  |-  ( w  =  suc  z  -> 
( suc  y  =  w 
<->  suc  y  =  suc  z ) )
4846, 47imbi12d 233 . . . . . . . . . . . 12  |-  ( w  =  suc  z  -> 
( ( suc  y  ~~  w  ->  suc  y  =  w )  <->  ( suc  y  ~~  suc  z  ->  suc  y  =  suc  z ) ) )
4948biimprcd 159 . . . . . . . . . . 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 2584 . . . . . . . . 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 712 . . . . . . . 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 114 . . . . . . 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 69 . . . . . 6  |-  ( y  e.  om  ->  ( A. z  e.  om  ( y  ~~  z  ->  y  =  z )  ->  ( w  e. 
om  ->  ( suc  y  ~~  w  ->  suc  y  =  w ) ) ) )
5554ralrimdv 2549 . . . . 5  |-  ( y  e.  om  ->  ( A. z  e.  om  ( y  ~~  z  ->  y  =  z )  ->  A. w  e.  om  ( suc  y  ~~  w  ->  suc  y  =  w ) ) )
56 breq2 3993 . . . . . . 7  |-  ( w  =  z  ->  ( suc  y  ~~  w  <->  suc  y  ~~  z ) )
57 eqeq2 2180 . . . . . . 7  |-  ( w  =  z  ->  ( suc  y  =  w  <->  suc  y  =  z ) )
5856, 57imbi12d 233 . . . . . 6  |-  ( w  =  z  ->  (
( suc  y  ~~  w  ->  suc  y  =  w )  <->  ( suc  y  ~~  z  ->  suc  y  =  z )
) )
5958cbvralv 2696 . . . . 5  |-  ( A. w  e.  om  ( suc  y  ~~  w  ->  suc  y  =  w
)  <->  A. z  e.  om  ( suc  y  ~~  z  ->  suc  y  =  z ) )
6055, 59syl6ib 160 . . . 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 4584 . . 3  |-  ( A  e.  om  ->  A. z  e.  om  ( A  ~~  z  ->  A  =  z ) )
62 breq2 3993 . . . . 5  |-  ( z  =  B  ->  ( A  ~~  z  <->  A  ~~  B ) )
63 eqeq2 2180 . . . . 5  |-  ( z  =  B  ->  ( A  =  z  <->  A  =  B ) )
6462, 63imbi12d 233 . . . 4  |-  ( z  =  B  ->  (
( A  ~~  z  ->  A  =  z )  <-> 
( A  ~~  B  ->  A  =  B ) ) )
6564rspcv 2830 . . 3  |-  ( B  e.  om  ->  ( A. z  e.  om  ( A  ~~  z  ->  A  =  z )  ->  ( A  ~~  B  ->  A  =  B ) ) )
6661, 65mpan9 279 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  ~~  B  ->  A  =  B ) )
67 eqeng 6744 . . 3  |-  ( A  e.  om  ->  ( A  =  B  ->  A 
~~  B ) )
6867adantr 274 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  =  B  ->  A  ~~  B
) )
6966, 68impbid 128 1  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  ~~  B  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 703    = wceq 1348    e. wcel 2141   A.wral 2448   E.wrex 2449   (/)c0 3414   class class class wbr 3989   suc csuc 4350   omcom 4574    ~~ cen 6716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-er 6513  df-en 6719
This theorem is referenced by:  findcard2  6867  findcard2s  6868  unsnfidcex  6897  unsnfidcel  6898  exmidonfinlem  7170  hashen  10718  hashunlem  10739
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