Theorem bdop 15521

Description: The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.)

Assertion

Ref	Expression
bdop	⊢ BOUNDED ⟨𝑥, 𝑦⟩

Proof of Theorem bdop

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdvsn 15520	. . . 4 ⊢ BOUNDED 𝑧 = {𝑥}
2		bdcpr 15517	. . . . . . 7 ⊢ BOUNDED {𝑥, 𝑦}
3	2	bdss 15510	. . . . . 6 ⊢ BOUNDED 𝑧 ⊆ {𝑥, 𝑦}
4		ax-bdel 15467	. . . . . . . 8 ⊢ BOUNDED 𝑥 ∈ 𝑧
5		ax-bdel 15467	. . . . . . . 8 ⊢ BOUNDED 𝑦 ∈ 𝑧
6	4, 5	ax-bdan 15461	. . . . . . 7 ⊢ BOUNDED (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑧)
7		vex 2766	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
8	7	prid1 3728	. . . . . . . . . 10 ⊢ 𝑥 ∈ {𝑥, 𝑦}
9		ssel 3177	. . . . . . . . . 10 ⊢ ({𝑥, 𝑦} ⊆ 𝑧 → (𝑥 ∈ {𝑥, 𝑦} → 𝑥 ∈ 𝑧))
10	8, 9	mpi 15	. . . . . . . . 9 ⊢ ({𝑥, 𝑦} ⊆ 𝑧 → 𝑥 ∈ 𝑧)
11		vex 2766	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
12	11	prid2 3729	. . . . . . . . . 10 ⊢ 𝑦 ∈ {𝑥, 𝑦}
13		ssel 3177	. . . . . . . . . 10 ⊢ ({𝑥, 𝑦} ⊆ 𝑧 → (𝑦 ∈ {𝑥, 𝑦} → 𝑦 ∈ 𝑧))
14	12, 13	mpi 15	. . . . . . . . 9 ⊢ ({𝑥, 𝑦} ⊆ 𝑧 → 𝑦 ∈ 𝑧)
15	10, 14	jca 306	. . . . . . . 8 ⊢ ({𝑥, 𝑦} ⊆ 𝑧 → (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑧))
16		prssi 3780	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑧) → {𝑥, 𝑦} ⊆ 𝑧)
17	15, 16	impbii 126	. . . . . . 7 ⊢ ({𝑥, 𝑦} ⊆ 𝑧 ↔ (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑧))
18	6, 17	bd0r 15471	. . . . . 6 ⊢ BOUNDED {𝑥, 𝑦} ⊆ 𝑧
19	3, 18	ax-bdan 15461	. . . . 5 ⊢ BOUNDED (𝑧 ⊆ {𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑧)
20		eqss 3198	. . . . 5 ⊢ (𝑧 = {𝑥, 𝑦} ↔ (𝑧 ⊆ {𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑧))
21	19, 20	bd0r 15471	. . . 4 ⊢ BOUNDED 𝑧 = {𝑥, 𝑦}
22	1, 21	ax-bdor 15462	. . 3 ⊢ BOUNDED (𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦})
23		vex 2766	. . . 4 ⊢ 𝑧 ∈ V
24	23, 7, 11	elop 4264	. . 3 ⊢ (𝑧 ∈ ⟨𝑥, 𝑦⟩ ↔ (𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦}))
25	22, 24	bd0r 15471	. 2 ⊢ BOUNDED 𝑧 ∈ ⟨𝑥, 𝑦⟩
26	25	bdelir 15493	1 ⊢ BOUNDED ⟨𝑥, 𝑦⟩

Syntax hints:   ∧ wa 104   ∨ wo 709   = wceq 1364   ∈ wcel 2167   ⊆ wss 3157  {csn 3622  {cpr 3623  ⟨cop 3625  BOUNDED wbdc 15486

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-3an 982  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-pr 3629  df-op 3631  df-bdc 15487

This theorem is referenced by: (None)