Theorem phplem3g 7110

Description: A natural number is equinumerous to its successor minus any element of the successor. Version of phplem3 7108 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 2295	. . . . 5 |- ( b = B -> ( b e. suc A <-> B e. suc A ) )
2	1	anbi2d 464	. . . 4 |- ( b = B -> ( ( A e. om /\ b e. suc A ) <-> ( A e. om /\ B e. suc A ) ) )
3		sneq 3700	. . . . . 6 |- ( b = B -> { b } = { B } )
4	3	difeq2d 3337	. . . . 5 |- ( b = B -> ( suc A \ { b } ) = ( suc A \ { B } ) )
5	4	breq2d 4121	. . . 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 2295	. . . . . . 7 |- ( a = A -> ( a e. om <-> A e. om ) )
8		suceq 4523	. . . . . . . 8 |- ( a = A -> suc a = suc A )
9	8	eleq2d 2302	. . . . . . 7 |- ( a = A -> ( b e. suc a <-> b e. suc A ) )
10	7, 9	anbi12d 473	. . . . . 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 3336	. . . . . . 7 |- ( a = A -> ( suc a \ { b } ) = ( suc A \ { b } ) )
13	11, 12	breq12d 4122	. . . . . 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 2816	. . . . . 6 |- a e. _V
16		vex 2816	. . . . . 6 |- b e. _V
17	15, 16	phplem3 7108	. . . . 5 |- ( ( a e. om /\ b e. suc a ) -> a ~~ ( suc a \ { b } ) )
18	14, 17	vtoclg 2875	. . . 4 |- ( A e. om -> ( ( A e. om /\ b e. suc A ) -> A ~~ ( suc A \ { b } ) ) )
19	18	anabsi5 581	. . 3 |- ( ( A e. om /\ b e. suc A ) -> A ~~ ( suc A \ { b } ) )
20	6, 19	vtoclg 2875	. 2 |- ( B e. suc A -> ( ( A e. om /\ B e. suc A ) -> A ~~ ( suc A \ { B } ) ) )
21	20	anabsi7 583	1 |- ( ( A e. om /\ B e. suc A ) -> A ~~ ( suc A \ { B } ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   \ cdif 3208   {csn 3689   class class class wbr 4109   suc csuc 4486   omcom 4712   ~~ cen 6973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-en 6976

This theorem is referenced by:  phplem4dom  7116  phpm  7120  phplem4on  7122