Theorem nndifsnid 6753

Description: If we remove a single element from a natural number then put it back in, we end up with the original natural number. This strengthens difsnss 3845 from subset to equality but the proof relies on equality being decidable. (Contributed by Jim Kingdon, 31-Aug-2021.)

Assertion

Ref	Expression
nndifsnid	|- ( ( A e. om /\ B e. A ) -> ( ( A \ { B } ) u. { B } ) = A )

Proof of Theorem nndifsnid

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elnn 4733	. . . . . 6 |- ( ( x e. A /\ A e. om ) -> x e. om )
2	1	expcom 116	. . . . 5 |- ( A e. om -> ( x e. A -> x e. om ) )
3		elnn 4733	. . . . . 6 |- ( ( y e. A /\ A e. om ) -> y e. om )
4	3	expcom 116	. . . . 5 |- ( A e. om -> ( y e. A -> y e. om ) )
5	2, 4	anim12d 335	. . . 4 |- ( A e. om -> ( ( x e. A /\ y e. A ) -> ( x e. om /\ y e. om ) ) )
6		nndceq 6745	. . . 4 |- ( ( x e. om /\ y e. om ) -> DECID x = y )
7	5, 6	syl6 33	. . 3 |- ( A e. om -> ( ( x e. A /\ y e. A ) -> DECID x = y ) )
8	7	ralrimivv 2625	. 2 |- ( A e. om -> A. x e. A A. y e. A DECID x = y )
9		dcdifsnid 6750	. 2 |- ( ( A. x e. A A. y e. A DECID x = y /\ B e. A ) -> ( ( A \ { B } ) u. { B } ) = A )
10	8, 9	sylan 283	1 |- ( ( A e. om /\ B e. A ) -> ( ( A \ { B } ) u. { B } ) = A )

Syntax hints:   -> wi 4   /\ wa 104  DECID wdc 842   = wceq 1398   e. wcel 2205   A.wral 2522   \ cdif 3211   u. cun 3212   {csn 3694   omcom 4717

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-uni 3920  df-int 3955  df-tr 4214  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718

This theorem is referenced by:  phplem2  7120