Theorem bdeqsuc 14636

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 14635	. . . 4 ⊢ BOUNDED suc 𝑦
2	1	bdss 14619	. . 3 ⊢ BOUNDED 𝑥 ⊆ suc 𝑦
3		bdcv 14603	. . . . . . 7 ⊢ BOUNDED 𝑥
4	3	bdss 14619	. . . . . 6 ⊢ BOUNDED 𝑦 ⊆ 𝑥
5	3	bdsnss 14628	. . . . . 6 ⊢ BOUNDED {𝑦} ⊆ 𝑥
6	4, 5	ax-bdan 14570	. . . . 5 ⊢ BOUNDED (𝑦 ⊆ 𝑥 ∧ {𝑦} ⊆ 𝑥)
7		unss 3310	. . . . 5 ⊢ ((𝑦 ⊆ 𝑥 ∧ {𝑦} ⊆ 𝑥) ↔ (𝑦 ∪ {𝑦}) ⊆ 𝑥)
8	6, 7	bd0 14579	. . . 4 ⊢ BOUNDED (𝑦 ∪ {𝑦}) ⊆ 𝑥
9		df-suc 4372	. . . . 5 ⊢ suc 𝑦 = (𝑦 ∪ {𝑦})
10	9	sseq1i 3182	. . . 4 ⊢ (suc 𝑦 ⊆ 𝑥 ↔ (𝑦 ∪ {𝑦}) ⊆ 𝑥)
11	8, 10	bd0r 14580	. . 3 ⊢ BOUNDED suc 𝑦 ⊆ 𝑥
12	2, 11	ax-bdan 14570	. 2 ⊢ BOUNDED (𝑥 ⊆ suc 𝑦 ∧ suc 𝑦 ⊆ 𝑥)
13		eqss 3171	. 2 ⊢ (𝑥 = suc 𝑦 ↔ (𝑥 ⊆ suc 𝑦 ∧ suc 𝑦 ⊆ 𝑥))
14	12, 13	bd0r 14580	1 ⊢ BOUNDED 𝑥 = suc 𝑦

Syntax hints:   ∧ wa 104   = wceq 1353   ∪ cun 3128   ⊆ wss 3130  {csn 3593  suc csuc 4366  BOUNDED wbd 14567

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-bd0 14568  ax-bdan 14570  ax-bdor 14571  ax-bdal 14573  ax-bdeq 14575  ax-bdel 14576  ax-bdsb 14577

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-sn 3599  df-suc 4372  df-bdc 14596

This theorem is referenced by:  bj-bdsucel  14637  bj-nn0suc0  14705