Theorem bdeqsuc 15527

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 15526	. . . 4 ⊢ BOUNDED suc 𝑦
2	1	bdss 15510	. . 3 ⊢ BOUNDED 𝑥 ⊆ suc 𝑦
3		bdcv 15494	. . . . . . 7 ⊢ BOUNDED 𝑥
4	3	bdss 15510	. . . . . 6 ⊢ BOUNDED 𝑦 ⊆ 𝑥
5	3	bdsnss 15519	. . . . . 6 ⊢ BOUNDED {𝑦} ⊆ 𝑥
6	4, 5	ax-bdan 15461	. . . . 5 ⊢ BOUNDED (𝑦 ⊆ 𝑥 ∧ {𝑦} ⊆ 𝑥)
7		unss 3337	. . . . 5 ⊢ ((𝑦 ⊆ 𝑥 ∧ {𝑦} ⊆ 𝑥) ↔ (𝑦 ∪ {𝑦}) ⊆ 𝑥)
8	6, 7	bd0 15470	. . . 4 ⊢ BOUNDED (𝑦 ∪ {𝑦}) ⊆ 𝑥
9		df-suc 4406	. . . . 5 ⊢ suc 𝑦 = (𝑦 ∪ {𝑦})
10	9	sseq1i 3209	. . . 4 ⊢ (suc 𝑦 ⊆ 𝑥 ↔ (𝑦 ∪ {𝑦}) ⊆ 𝑥)
11	8, 10	bd0r 15471	. . 3 ⊢ BOUNDED suc 𝑦 ⊆ 𝑥
12	2, 11	ax-bdan 15461	. 2 ⊢ BOUNDED (𝑥 ⊆ suc 𝑦 ∧ suc 𝑦 ⊆ 𝑥)
13		eqss 3198	. 2 ⊢ (𝑥 = suc 𝑦 ↔ (𝑥 ⊆ suc 𝑦 ∧ suc 𝑦 ⊆ 𝑥))
14	12, 13	bd0r 15471	1 ⊢ BOUNDED 𝑥 = suc 𝑦

Syntax hints:   ∧ wa 104   = wceq 1364   ∪ cun 3155   ⊆ wss 3157  {csn 3622  suc csuc 4400  BOUNDED wbd 15458

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-bd0 15459  ax-bdan 15461  ax-bdor 15462  ax-bdal 15464  ax-bdeq 15466  ax-bdel 15467  ax-bdsb 15468

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-suc 4406  df-bdc 15487

This theorem is referenced by:  bj-bdsucel  15528  bj-nn0suc0  15596