Theorem phplem3g 7157

Description: A natural number is equinumerous to its successor minus any element of the successor. Version of phplem3 7155 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.

Step	Hyp	Ref	Expression
1		eleq1 2301	. . . . 5 |- ( b = B -> ( b e. suc A <-> B e. suc A ) )
2	1	anbi2d 468	. . . 4 |- ( b = B -> ( ( A e. om /\ b e. suc A ) <-> ( A e. om /\ B e. suc A ) ) )
3		sneq 3720	. . . . . 6 |- ( b = B -> { b } = { B } )
4	3	difeq2d 3347	. . . . 5 |- ( b = B -> ( suc A \ { b } ) = ( suc A \ { B } ) )
5	4	breq2d 4142	. . . 4 |- ( b = B -> ( A ~~ ( suc A \ { b } ) <-> A ~~ ( suc A \ { B } ) ) )
6	2, 5	imbi12d 234	. . 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 2301	. . . . . . 7 |- ( a = A -> ( a e. om <-> A e. om ) )
8		suceq 4547	. . . . . . . 8 |- ( a = A -> suc a = suc A )
9	8	eleq2d 2308	. . . . . . 7 |- ( a = A -> ( b e. suc a <-> b e. suc A ) )
10	7, 9	anbi12d 477	. . . . . 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 )
12	8	difeq1d 3346	. . . . . . 7 |- ( a = A -> ( suc a \ { b } ) = ( suc A \ { b } ) )
13	11, 12	breq12d 4143	. . . . . 6 |- ( a = A -> ( a ~~ ( suc a \ { b } ) <-> A ~~ ( suc A \ { b } ) ) )
14	10, 13	imbi12d 234	. . . . 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 2824	. . . . . 6 |- a e. _V
16		vex 2824	. . . . . 6 |- b e. _V
17	15, 16	phplem3 7155	. . . . 5 |- ( ( a e. om /\ b e. suc a ) -> a ~~ ( suc a \ { b } ) )
18	14, 17	vtoclg 2883	. . . 4 |- ( A e. om -> ( ( A e. om /\ b e. suc A ) -> A ~~ ( suc A \ { b } ) ) )
19	18	anabsi5 585	. . 3 |- ( ( A e. om /\ b e. suc A ) -> A ~~ ( suc A \ { b } ) )
20	6, 19	vtoclg 2883	. 2 |- ( B e. suc A -> ( ( A e. om /\ B e. suc A ) -> A ~~ ( suc A \ { B } ) ) )
21	20	anabsi7 587	1 |- ( ( A e. om /\ B e. suc A ) -> A ~~ ( suc A \ { B } ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   \ cdif 3217   {csn 3709   class class class wbr 4130   suc csuc 4510   omcom 4737   ~~ cen 7020

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-en 7023

This theorem is used by:  phplem4dom  7163  phpm  7167  phplem4on  7169