Theorem bdeqsuc 15750

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 15749	. . . 4 ⊢ BOUNDED suc 𝑦
2	1	bdss 15733	. . 3 ⊢ BOUNDED 𝑥 ⊆ suc 𝑦
3		bdcv 15717	. . . . . . 7 ⊢ BOUNDED 𝑥
4	3	bdss 15733	. . . . . 6 ⊢ BOUNDED 𝑦 ⊆ 𝑥
5	3	bdsnss 15742	. . . . . 6 ⊢ BOUNDED {𝑦} ⊆ 𝑥
6	4, 5	ax-bdan 15684	. . . . 5 ⊢ BOUNDED (𝑦 ⊆ 𝑥 ∧ {𝑦} ⊆ 𝑥)
7		unss 3346	. . . . 5 ⊢ ((𝑦 ⊆ 𝑥 ∧ {𝑦} ⊆ 𝑥) ↔ (𝑦 ∪ {𝑦}) ⊆ 𝑥)
8	6, 7	bd0 15693	. . . 4 ⊢ BOUNDED (𝑦 ∪ {𝑦}) ⊆ 𝑥
9		df-suc 4417	. . . . 5 ⊢ suc 𝑦 = (𝑦 ∪ {𝑦})
10	9	sseq1i 3218	. . . 4 ⊢ (suc 𝑦 ⊆ 𝑥 ↔ (𝑦 ∪ {𝑦}) ⊆ 𝑥)
11	8, 10	bd0r 15694	. . 3 ⊢ BOUNDED suc 𝑦 ⊆ 𝑥
12	2, 11	ax-bdan 15684	. 2 ⊢ BOUNDED (𝑥 ⊆ suc 𝑦 ∧ suc 𝑦 ⊆ 𝑥)
13		eqss 3207	. 2 ⊢ (𝑥 = suc 𝑦 ↔ (𝑥 ⊆ suc 𝑦 ∧ suc 𝑦 ⊆ 𝑥))
14	12, 13	bd0r 15694	1 ⊢ BOUNDED 𝑥 = suc 𝑦

Syntax hints:   ∧ wa 104   = wceq 1372   ∪ cun 3163   ⊆ wss 3165  {csn 3632  suc csuc 4411  BOUNDED wbd 15681

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186  ax-bd0 15682  ax-bdan 15684  ax-bdor 15685  ax-bdal 15687  ax-bdeq 15689  ax-bdel 15690  ax-bdsb 15691

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-sn 3638  df-suc 4417  df-bdc 15710

This theorem is referenced by:  bj-bdsucel  15751  bj-nn0suc0  15819