Theorem bdeqsuc 10001

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 10000	. . . 4 |- BOUNDED suc y
2	1	bdss 9984	. . 3 |- BOUNDED x C_ suc y
3		bdcv 9968	. . . . . . 7 |- BOUNDED x
4	3	bdss 9984	. . . . . 6 |- BOUNDED y C_ x
5	3	bdsnss 9993	. . . . . 6 |- BOUNDED { y } C_ x
6	4, 5	ax-bdan 9935	. . . . 5 |- BOUNDED ( y C_ x /\ { y } C_ x )
7		unss 3117	. . . . 5 |- ( ( y C_ x /\ { y } C_ x ) <-> ( y u. { y } ) C_ x )
8	6, 7	bd0 9944	. . . 4 |- BOUNDED ( y u. { y } ) C_ x
9		df-suc 4108	. . . . 5 |- suc y = ( y u. { y } )
10	9	sseq1i 2969	. . . 4 |- ( suc y C_ x <-> ( y u. { y } ) C_ x )
11	8, 10	bd0r 9945	. . 3 |- BOUNDED suc y C_ x
12	2, 11	ax-bdan 9935	. 2 |- BOUNDED ( x C_ suc y /\ suc y C_ x )
13		eqss 2960	. 2 |- ( x = suc y <-> ( x C_ suc y /\ suc y C_ x ) )
14	12, 13	bd0r 9945	1 |- BOUNDED x = suc y

Syntax hints:   /\ wa 97   = wceq 1243   u. cun 2915   C_ wss 2917   {csn 3375   suc csuc 4102  BOUNDED wbd 9932

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-bd0 9933  ax-bdan 9935  ax-bdor 9936  ax-bdal 9938  ax-bdeq 9940  ax-bdel 9941  ax-bdsb 9942

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-suc 4108  df-bdc 9961

This theorem is referenced by:  bj-bdsucel  10002  bj-nn0suc0  10075