Theorem phplem3g 6834

Description: A natural number is equinumerous to its successor minus any element of the successor. Version of phplem3 6832 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 2233	. . . . 5 |- ( b = B -> ( b e. suc A <-> B e. suc A ) )
2	1	anbi2d 461	. . . 4 |- ( b = B -> ( ( A e. om /\ b e. suc A ) <-> ( A e. om /\ B e. suc A ) ) )
3		sneq 3594	. . . . . 6 |- ( b = B -> { b } = { B } )
4	3	difeq2d 3245	. . . . 5 |- ( b = B -> ( suc A \ { b } ) = ( suc A \ { B } ) )
5	4	breq2d 4001	. . . 4 |- ( b = B -> ( A ~~ ( suc A \ { b } ) <-> A ~~ ( suc A \ { B } ) ) )
6	2, 5	imbi12d 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 2233	. . . . . . 7 |- ( a = A -> ( a e. om <-> A e. om ) )
8		suceq 4387	. . . . . . . 8 |- ( a = A -> suc a = suc A )
9	8	eleq2d 2240	. . . . . . 7 |- ( a = A -> ( b e. suc a <-> b e. suc A ) )
10	7, 9	anbi12d 470	. . . . . 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 3244	. . . . . . 7 |- ( a = A -> ( suc a \ { b } ) = ( suc A \ { b } ) )
13	11, 12	breq12d 4002	. . . . . 6 |- ( a = A -> ( a ~~ ( suc a \ { b } ) <-> A ~~ ( suc A \ { b } ) ) )
14	10, 13	imbi12d 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 2733	. . . . . 6 |- a e. _V
16		vex 2733	. . . . . 6 |- b e. _V
17	15, 16	phplem3 6832	. . . . 5 |- ( ( a e. om /\ b e. suc a ) -> a ~~ ( suc a \ { b } ) )
18	14, 17	vtoclg 2790	. . . 4 |- ( A e. om -> ( ( A e. om /\ b e. suc A ) -> A ~~ ( suc A \ { b } ) ) )
19	18	anabsi5 574	. . 3 |- ( ( A e. om /\ b e. suc A ) -> A ~~ ( suc A \ { b } ) )
20	6, 19	vtoclg 2790	. 2 |- ( B e. suc A -> ( ( A e. om /\ B e. suc A ) -> A ~~ ( suc A \ { B } ) ) )
21	20	anabsi7 576	1 |- ( ( A e. om /\ B e. suc A ) -> A ~~ ( suc A \ { B } ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   \ cdif 3118   {csn 3583   class class class wbr 3989   suc csuc 4350   omcom 4574   ~~ cen 6716

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-en 6719

This theorem is referenced by:  phplem4dom  6840  phpm  6843  phplem4on  6845