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Theorem phplem3g 6834
Description: A natural number is equinumerous to its successor minus any element of the successor. Version of phplem3 6832 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 2233 . . . . 5  |-  ( b  =  B  ->  (
b  e.  suc  A  <->  B  e.  suc  A ) )
21anbi2d 461 . . . 4  |-  ( b  =  B  ->  (
( A  e.  om  /\  b  e.  suc  A
)  <->  ( A  e. 
om  /\  B  e.  suc  A ) ) )
3 sneq 3594 . . . . . 6  |-  ( b  =  B  ->  { b }  =  { B } )
43difeq2d 3245 . . . . 5  |-  ( b  =  B  ->  ( suc  A  \  { b } )  =  ( suc  A  \  { B } ) )
54breq2d 4001 . . . 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 2233 . . . . . . 7  |-  ( a  =  A  ->  (
a  e.  om  <->  A  e.  om ) )
8 suceq 4387 . . . . . . . 8  |-  ( a  =  A  ->  suc  a  =  suc  A )
98eleq2d 2240 . . . . . . 7  |-  ( a  =  A  ->  (
b  e.  suc  a  <->  b  e.  suc  A ) )
107, 9anbi12d 470 . . . . . 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 3244 . . . . . . 7  |-  ( a  =  A  ->  ( suc  a  \  { b } )  =  ( suc  A  \  {
b } ) )
1311, 12breq12d 4002 . . . . . 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 2733 . . . . . 6  |-  a  e. 
_V
16 vex 2733 . . . . . 6  |-  b  e. 
_V
1715, 16phplem3 6832 . . . . 5  |-  ( ( a  e.  om  /\  b  e.  suc  a )  ->  a  ~~  ( suc  a  \  { b } ) )
1814, 17vtoclg 2790 . . . 4  |-  ( A  e.  om  ->  (
( A  e.  om  /\  b  e.  suc  A
)  ->  A  ~~  ( suc  A  \  {
b } ) ) )
1918anabsi5 574 . . 3  |-  ( ( A  e.  om  /\  b  e.  suc  A )  ->  A  ~~  ( suc  A  \  { b } ) )
206, 19vtoclg 2790 . 2  |-  ( B  e.  suc  A  -> 
( ( A  e. 
om  /\  B  e.  suc  A )  ->  A  ~~  ( suc  A  \  { B } ) ) )
2120anabsi7 576 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 1348    e. wcel 2141    \ cdif 3118   {csn 3583   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-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-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-en 6719
This theorem is referenced by:  phplem4dom  6840  phpm  6843  phplem4on  6845
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