Theorem phplem3g 6319

Description: A natural number is equinumerous to its successor minus any element of the successor. Version of phplem3 6317 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 2100	. . . . 5 |- ( b = B -> ( b e. suc A <-> B e. suc A ) )
2	1	anbi2d 437	. . . 4 |- ( b = B -> ( ( A e. om /\ b e. suc A ) <-> ( A e. om /\ B e. suc A ) ) )
3		sneq 3386	. . . . . 6 |- ( b = B -> { b } = { B } )
4	3	difeq2d 3062	. . . . 5 |- ( b = B -> ( suc A \ { b } ) = ( suc A \ { B } ) )
5	4	breq2d 3776	. . . 4 |- ( b = B -> ( A ~~ ( suc A \ { b } ) <-> A ~~ ( suc A \ { B } ) ) )
6	2, 5	imbi12d 223	. . 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 2100	. . . . . . 7 |- ( a = A -> ( a e. om <-> A e. om ) )
8		suceq 4139	. . . . . . . 8 |- ( a = A -> suc a = suc A )
9	8	eleq2d 2107	. . . . . . 7 |- ( a = A -> ( b e. suc a <-> b e. suc A ) )
10	7, 9	anbi12d 442	. . . . . 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 3061	. . . . . . 7 |- ( a = A -> ( suc a \ { b } ) = ( suc A \ { b } ) )
13	11, 12	breq12d 3777	. . . . . 6 |- ( a = A -> ( a ~~ ( suc a \ { b } ) <-> A ~~ ( suc A \ { b } ) ) )
14	10, 13	imbi12d 223	. . . . 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 2560	. . . . . 6 |- a e. _V
16		vex 2560	. . . . . 6 |- b e. _V
17	15, 16	phplem3 6317	. . . . 5 |- ( ( a e. om /\ b e. suc a ) -> a ~~ ( suc a \ { b } ) )
18	14, 17	vtoclg 2613	. . . 4 |- ( A e. om -> ( ( A e. om /\ b e. suc A ) -> A ~~ ( suc A \ { b } ) ) )
19	18	anabsi5 513	. . 3 |- ( ( A e. om /\ b e. suc A ) -> A ~~ ( suc A \ { b } ) )
20	6, 19	vtoclg 2613	. 2 |- ( B e. suc A -> ( ( A e. om /\ B e. suc A ) -> A ~~ ( suc A \ { B } ) ) )
21	20	anabsi7 515	1 |- ( ( A e. om /\ B e. suc A ) -> A ~~ ( suc A \ { B } ) )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   \ cdif 2914   {csn 3375   class class class wbr 3764   suc csuc 4102   omcom 4313   ~~ cen 6219

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-br 3765  df-opab 3819  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-en 6222

This theorem is referenced by:  phplem4dom  6324  phpm  6327  phplem4on  6329