Theorem bdeqsuc 15821

Description: Boundedness of the formula expressing that a setvar is equal to the successor of another. (Contributed by BJ, 21-Nov-2019.)

Assertion

Ref	Expression
bdeqsuc	|- BOUNDED x = suc y

Distinct variable group:   x, y

Proof of Theorem bdeqsuc

Step	Hyp	Ref	Expression
1		bdcsuc 15820	. . . 4 |- BOUNDED suc y
2	1	bdss 15804	. . 3 |- BOUNDED x C_ suc y
3		bdcv 15788	. . . . . . 7 |- BOUNDED x
4	3	bdss 15804	. . . . . 6 |- BOUNDED y C_ x
5	3	bdsnss 15813	. . . . . 6 |- BOUNDED { y } C_ x
6	4, 5	ax-bdan 15755	. . . . 5 |- BOUNDED ( y C_ x /\ { y } C_ x )
7		unss 3347	. . . . 5 |- ( ( y C_ x /\ { y } C_ x ) <-> ( y u. { y } ) C_ x )
8	6, 7	bd0 15764	. . . 4 |- BOUNDED ( y u. { y } ) C_ x
9		df-suc 4418	. . . . 5 |- suc y = ( y u. { y } )
10	9	sseq1i 3219	. . . 4 |- ( suc y C_ x <-> ( y u. { y } ) C_ x )
11	8, 10	bd0r 15765	. . 3 |- BOUNDED suc y C_ x
12	2, 11	ax-bdan 15755	. 2 |- BOUNDED ( x C_ suc y /\ suc y C_ x )
13		eqss 3208	. 2 |- ( x = suc y <-> ( x C_ suc y /\ suc y C_ x ) )
14	12, 13	bd0r 15765	1 |- BOUNDED x = suc y

Syntax hints:   /\ wa 104   = wceq 1373   u. cun 3164   C_ wss 3166   {csn 3633   suc csuc 4412  BOUNDED wbd 15752

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-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-bd0 15753  ax-bdan 15755  ax-bdor 15756  ax-bdal 15758  ax-bdeq 15760  ax-bdel 15761  ax-bdsb 15762

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-suc 4418  df-bdc 15781

This theorem is referenced by:  bj-bdsucel  15822  bj-nn0suc0  15890