Theorem bdeqsuc 16412

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 16411	. . . 4 ⊢ BOUNDED suc 𝑦
2	1	bdss 16395	. . 3 ⊢ BOUNDED 𝑥 ⊆ suc 𝑦
3		bdcv 16379	. . . . . . 7 ⊢ BOUNDED 𝑥
4	3	bdss 16395	. . . . . 6 ⊢ BOUNDED 𝑦 ⊆ 𝑥
5	3	bdsnss 16404	. . . . . 6 ⊢ BOUNDED {𝑦} ⊆ 𝑥
6	4, 5	ax-bdan 16346	. . . . 5 ⊢ BOUNDED (𝑦 ⊆ 𝑥 ∧ {𝑦} ⊆ 𝑥)
7		unss 3379	. . . . 5 ⊢ ((𝑦 ⊆ 𝑥 ∧ {𝑦} ⊆ 𝑥) ↔ (𝑦 ∪ {𝑦}) ⊆ 𝑥)
8	6, 7	bd0 16355	. . . 4 ⊢ BOUNDED (𝑦 ∪ {𝑦}) ⊆ 𝑥
9		df-suc 4466	. . . . 5 ⊢ suc 𝑦 = (𝑦 ∪ {𝑦})
10	9	sseq1i 3251	. . . 4 ⊢ (suc 𝑦 ⊆ 𝑥 ↔ (𝑦 ∪ {𝑦}) ⊆ 𝑥)
11	8, 10	bd0r 16356	. . 3 ⊢ BOUNDED suc 𝑦 ⊆ 𝑥
12	2, 11	ax-bdan 16346	. 2 ⊢ BOUNDED (𝑥 ⊆ suc 𝑦 ∧ suc 𝑦 ⊆ 𝑥)
13		eqss 3240	. 2 ⊢ (𝑥 = suc 𝑦 ↔ (𝑥 ⊆ suc 𝑦 ∧ suc 𝑦 ⊆ 𝑥))
14	12, 13	bd0r 16356	1 ⊢ BOUNDED 𝑥 = suc 𝑦

Syntax hints:   ∧ wa 104   = wceq 1395   ∪ cun 3196   ⊆ wss 3198  {csn 3667  suc csuc 4460  BOUNDED wbd 16343

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-bd0 16344  ax-bdan 16346  ax-bdor 16347  ax-bdal 16349  ax-bdeq 16351  ax-bdel 16352  ax-bdsb 16353

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-sn 3673  df-suc 4466  df-bdc 16372

This theorem is referenced by:  bj-bdsucel  16413  bj-nn0suc0  16481