Theorem nndifsnid 6718

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 3824 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 4710	. . . . . 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 4710	. . . . . 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 6710	. . . 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 2614	. 2 |- ( A e. om -> A. x e. A A. y e. A DECID x = y )
9		dcdifsnid 6715	. 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 2202   A.wral 2511   \ cdif 3198   u. cun 3199   {csn 3673   omcom 4694

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

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 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-uni 3899  df-int 3934  df-tr 4193  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695

This theorem is referenced by:  phplem2  7082