Theorem phplem3g 6912

Description: A natural number is equinumerous to its successor minus any element of the successor. Version of phplem3 6910 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 2256	. . . . 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 3629	. . . . . 6 |- ( b = B -> { b } = { B } )
4	3	difeq2d 3277	. . . . 5 |- ( b = B -> ( suc A \ { b } ) = ( suc A \ { B } ) )
5	4	breq2d 4041	. . . 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 2256	. . . . . . 7 |- ( a = A -> ( a e. om <-> A e. om ) )
8		suceq 4433	. . . . . . . 8 |- ( a = A -> suc a = suc A )
9	8	eleq2d 2263	. . . . . . 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 3276	. . . . . . 7 |- ( a = A -> ( suc a \ { b } ) = ( suc A \ { b } ) )
13	11, 12	breq12d 4042	. . . . . 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 2763	. . . . . 6 |- a e. _V
16		vex 2763	. . . . . 6 |- b e. _V
17	15, 16	phplem3 6910	. . . . 5 |- ( ( a e. om /\ b e. suc a ) -> a ~~ ( suc a \ { b } ) )
18	14, 17	vtoclg 2820	. . . 4 |- ( A e. om -> ( ( A e. om /\ b e. suc A ) -> A ~~ ( suc A \ { b } ) ) )
19	18	anabsi5 579	. . 3 |- ( ( A e. om /\ b e. suc A ) -> A ~~ ( suc A \ { b } ) )
20	6, 19	vtoclg 2820	. 2 |- ( B e. suc A -> ( ( A e. om /\ B e. suc A ) -> A ~~ ( suc A \ { B } ) ) )
21	20	anabsi7 581	1 |- ( ( A e. om /\ B e. suc A ) -> A ~~ ( suc A \ { B } ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   \ cdif 3150   {csn 3618   class class class wbr 4029   suc csuc 4396   omcom 4622   ~~ cen 6792

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-en 6795

This theorem is referenced by:  phplem4dom  6918  phpm  6921  phplem4on  6923