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Theorem phplem3g 6718
Description: A natural number is equinumerous to its successor minus any element of the successor. Version of phplem3 6716 with unnecessary hypotheses removed. (Contributed by Jim Kingdon, 1-Sep-2021.)
Assertion
Ref Expression
phplem3g  |-  ( ( A  e.  om  /\  B  e.  suc  A )  ->  A  ~~  ( suc  A  \  { B } ) )

Proof of Theorem phplem3g
Dummy variables  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2180 . . . . 5  |-  ( b  =  B  ->  (
b  e.  suc  A  <->  B  e.  suc  A ) )
21anbi2d 459 . . . 4  |-  ( b  =  B  ->  (
( A  e.  om  /\  b  e.  suc  A
)  <->  ( A  e. 
om  /\  B  e.  suc  A ) ) )
3 sneq 3508 . . . . . 6  |-  ( b  =  B  ->  { b }  =  { B } )
43difeq2d 3164 . . . . 5  |-  ( b  =  B  ->  ( suc  A  \  { b } )  =  ( suc  A  \  { B } ) )
54breq2d 3911 . . . 4  |-  ( b  =  B  ->  ( A  ~~  ( suc  A  \  { b } )  <-> 
A  ~~  ( suc  A 
\  { B }
) ) )
62, 5imbi12d 233 . . 3  |-  ( b  =  B  ->  (
( ( A  e. 
om  /\  b  e.  suc  A )  ->  A  ~~  ( suc  A  \  { b } ) )  <->  ( ( A  e.  om  /\  B  e.  suc  A )  ->  A  ~~  ( suc  A  \  { B } ) ) ) )
7 eleq1 2180 . . . . . . 7  |-  ( a  =  A  ->  (
a  e.  om  <->  A  e.  om ) )
8 suceq 4294 . . . . . . . 8  |-  ( a  =  A  ->  suc  a  =  suc  A )
98eleq2d 2187 . . . . . . 7  |-  ( a  =  A  ->  (
b  e.  suc  a  <->  b  e.  suc  A ) )
107, 9anbi12d 464 . . . . . 6  |-  ( a  =  A  ->  (
( a  e.  om  /\  b  e.  suc  a
)  <->  ( A  e. 
om  /\  b  e.  suc  A ) ) )
11 id 19 . . . . . . 7  |-  ( a  =  A  ->  a  =  A )
128difeq1d 3163 . . . . . . 7  |-  ( a  =  A  ->  ( suc  a  \  { b } )  =  ( suc  A  \  {
b } ) )
1311, 12breq12d 3912 . . . . . 6  |-  ( a  =  A  ->  (
a  ~~  ( suc  a  \  { b } )  <->  A  ~~  ( suc 
A  \  { b } ) ) )
1410, 13imbi12d 233 . . . . 5  |-  ( a  =  A  ->  (
( ( a  e. 
om  /\  b  e.  suc  a )  ->  a  ~~  ( suc  a  \  { b } ) )  <->  ( ( A  e.  om  /\  b  e.  suc  A )  ->  A  ~~  ( suc  A  \  { b } ) ) ) )
15 vex 2663 . . . . . 6  |-  a  e. 
_V
16 vex 2663 . . . . . 6  |-  b  e. 
_V
1715, 16phplem3 6716 . . . . 5  |-  ( ( a  e.  om  /\  b  e.  suc  a )  ->  a  ~~  ( suc  a  \  { b } ) )
1814, 17vtoclg 2720 . . . 4  |-  ( A  e.  om  ->  (
( A  e.  om  /\  b  e.  suc  A
)  ->  A  ~~  ( suc  A  \  {
b } ) ) )
1918anabsi5 553 . . 3  |-  ( ( A  e.  om  /\  b  e.  suc  A )  ->  A  ~~  ( suc  A  \  { b } ) )
206, 19vtoclg 2720 . 2  |-  ( B  e.  suc  A  -> 
( ( A  e. 
om  /\  B  e.  suc  A )  ->  A  ~~  ( suc  A  \  { B } ) ) )
2120anabsi7 555 1  |-  ( ( A  e.  om  /\  B  e.  suc  A )  ->  A  ~~  ( suc  A  \  { B } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465    \ cdif 3038   {csn 3497   class class class wbr 3899   suc csuc 4257   omcom 4474    ~~ cen 6600
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-en 6603
This theorem is referenced by:  phplem4dom  6724  phpm  6727  phplem4on  6729
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