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Theorem phplem3g 6802
Description: A natural number is equinumerous to its successor minus any element of the successor. Version of phplem3 6800 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 2220 . . . . 5  |-  ( b  =  B  ->  (
b  e.  suc  A  <->  B  e.  suc  A ) )
21anbi2d 460 . . . 4  |-  ( b  =  B  ->  (
( A  e.  om  /\  b  e.  suc  A
)  <->  ( A  e. 
om  /\  B  e.  suc  A ) ) )
3 sneq 3571 . . . . . 6  |-  ( b  =  B  ->  { b }  =  { B } )
43difeq2d 3225 . . . . 5  |-  ( b  =  B  ->  ( suc  A  \  { b } )  =  ( suc  A  \  { B } ) )
54breq2d 3978 . . . 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 2220 . . . . . . 7  |-  ( a  =  A  ->  (
a  e.  om  <->  A  e.  om ) )
8 suceq 4363 . . . . . . . 8  |-  ( a  =  A  ->  suc  a  =  suc  A )
98eleq2d 2227 . . . . . . 7  |-  ( a  =  A  ->  (
b  e.  suc  a  <->  b  e.  suc  A ) )
107, 9anbi12d 465 . . . . . 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 3224 . . . . . . 7  |-  ( a  =  A  ->  ( suc  a  \  { b } )  =  ( suc  A  \  {
b } ) )
1311, 12breq12d 3979 . . . . . 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 2715 . . . . . 6  |-  a  e. 
_V
16 vex 2715 . . . . . 6  |-  b  e. 
_V
1715, 16phplem3 6800 . . . . 5  |-  ( ( a  e.  om  /\  b  e.  suc  a )  ->  a  ~~  ( suc  a  \  { b } ) )
1814, 17vtoclg 2772 . . . 4  |-  ( A  e.  om  ->  (
( A  e.  om  /\  b  e.  suc  A
)  ->  A  ~~  ( suc  A  \  {
b } ) ) )
1918anabsi5 569 . . 3  |-  ( ( A  e.  om  /\  b  e.  suc  A )  ->  A  ~~  ( suc  A  \  { b } ) )
206, 19vtoclg 2772 . 2  |-  ( B  e.  suc  A  -> 
( ( A  e. 
om  /\  B  e.  suc  A )  ->  A  ~~  ( suc  A  \  { B } ) ) )
2120anabsi7 571 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 1335    e. wcel 2128    \ cdif 3099   {csn 3560   class class class wbr 3966   suc csuc 4326   omcom 4550    ~~ cen 6684
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-nul 4091  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-setind 4497  ax-iinf 4548
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-int 3809  df-br 3967  df-opab 4027  df-tr 4064  df-id 4254  df-iord 4327  df-on 4329  df-suc 4332  df-iom 4551  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-fun 5173  df-fn 5174  df-f 5175  df-f1 5176  df-fo 5177  df-f1o 5178  df-en 6687
This theorem is referenced by:  phplem4dom  6808  phpm  6811  phplem4on  6813
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