Theorem bdeqsuc 15373

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 15372	. . . 4 ⊢ BOUNDED suc 𝑦
2	1	bdss 15356	. . 3 ⊢ BOUNDED 𝑥 ⊆ suc 𝑦
3		bdcv 15340	. . . . . . 7 ⊢ BOUNDED 𝑥
4	3	bdss 15356	. . . . . 6 ⊢ BOUNDED 𝑦 ⊆ 𝑥
5	3	bdsnss 15365	. . . . . 6 ⊢ BOUNDED {𝑦} ⊆ 𝑥
6	4, 5	ax-bdan 15307	. . . . 5 ⊢ BOUNDED (𝑦 ⊆ 𝑥 ∧ {𝑦} ⊆ 𝑥)
7		unss 3333	. . . . 5 ⊢ ((𝑦 ⊆ 𝑥 ∧ {𝑦} ⊆ 𝑥) ↔ (𝑦 ∪ {𝑦}) ⊆ 𝑥)
8	6, 7	bd0 15316	. . . 4 ⊢ BOUNDED (𝑦 ∪ {𝑦}) ⊆ 𝑥
9		df-suc 4402	. . . . 5 ⊢ suc 𝑦 = (𝑦 ∪ {𝑦})
10	9	sseq1i 3205	. . . 4 ⊢ (suc 𝑦 ⊆ 𝑥 ↔ (𝑦 ∪ {𝑦}) ⊆ 𝑥)
11	8, 10	bd0r 15317	. . 3 ⊢ BOUNDED suc 𝑦 ⊆ 𝑥
12	2, 11	ax-bdan 15307	. 2 ⊢ BOUNDED (𝑥 ⊆ suc 𝑦 ∧ suc 𝑦 ⊆ 𝑥)
13		eqss 3194	. 2 ⊢ (𝑥 = suc 𝑦 ↔ (𝑥 ⊆ suc 𝑦 ∧ suc 𝑦 ⊆ 𝑥))
14	12, 13	bd0r 15317	1 ⊢ BOUNDED 𝑥 = suc 𝑦

Syntax hints:   ∧ wa 104   = wceq 1364   ∪ cun 3151   ⊆ wss 3153  {csn 3618  suc csuc 4396  BOUNDED wbd 15304

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-bd0 15305  ax-bdan 15307  ax-bdor 15308  ax-bdal 15310  ax-bdeq 15312  ax-bdel 15313  ax-bdsb 15314

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-suc 4402  df-bdc 15333

This theorem is referenced by:  bj-bdsucel  15374  bj-nn0suc0  15442