Theorem nndifsnid 6740

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 3840 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 4728	. . . . . 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 4728	. . . . . 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 6732	. . . 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 2623	. 2 |- ( A e. om -> A. x e. A A. y e. A DECID x = y )
9		dcdifsnid 6737	. 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 2203   A.wral 2520   \ cdif 3208   u. cun 3209   {csn 3689   omcom 4712

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-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  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-uni 3915  df-int 3950  df-tr 4209  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713

This theorem is referenced by:  phplem2  7107