Theorem bdop 16196

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 16195	. . . 4 ⊢ BOUNDED 𝑧 = {𝑥}
2		bdcpr 16192	. . . . . . 7 ⊢ BOUNDED {𝑥, 𝑦}
3	2	bdss 16185	. . . . . 6 ⊢ BOUNDED 𝑧 ⊆ {𝑥, 𝑦}
4		ax-bdel 16142	. . . . . . . 8 ⊢ BOUNDED 𝑥 ∈ 𝑧
5		ax-bdel 16142	. . . . . . . 8 ⊢ BOUNDED 𝑦 ∈ 𝑧
6	4, 5	ax-bdan 16136	. . . . . . 7 ⊢ BOUNDED (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑧)
7		vex 2802	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
8	7	prid1 3772	. . . . . . . . . 10 ⊢ 𝑥 ∈ {𝑥, 𝑦}
9		ssel 3218	. . . . . . . . . 10 ⊢ ({𝑥, 𝑦} ⊆ 𝑧 → (𝑥 ∈ {𝑥, 𝑦} → 𝑥 ∈ 𝑧))
10	8, 9	mpi 15	. . . . . . . . 9 ⊢ ({𝑥, 𝑦} ⊆ 𝑧 → 𝑥 ∈ 𝑧)
11		vex 2802	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
12	11	prid2 3773	. . . . . . . . . 10 ⊢ 𝑦 ∈ {𝑥, 𝑦}
13		ssel 3218	. . . . . . . . . 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 3825	. . . . . . . 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 16146	. . . . . 6 ⊢ BOUNDED {𝑥, 𝑦} ⊆ 𝑧
19	3, 18	ax-bdan 16136	. . . . 5 ⊢ BOUNDED (𝑧 ⊆ {𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑧)
20		eqss 3239	. . . . 5 ⊢ (𝑧 = {𝑥, 𝑦} ↔ (𝑧 ⊆ {𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑧))
21	19, 20	bd0r 16146	. . . 4 ⊢ BOUNDED 𝑧 = {𝑥, 𝑦}
22	1, 21	ax-bdor 16137	. . 3 ⊢ BOUNDED (𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦})
23		vex 2802	. . . 4 ⊢ 𝑧 ∈ V
24	23, 7, 11	elop 4316	. . 3 ⊢ (𝑧 ∈ ⟨𝑥, 𝑦⟩ ↔ (𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦}))
25	22, 24	bd0r 16146	. 2 ⊢ BOUNDED 𝑧 ∈ ⟨𝑥, 𝑦⟩
26	25	bdelir 16168	1 ⊢ BOUNDED ⟨𝑥, 𝑦⟩

Syntax hints:   ∧ wa 104   ∨ wo 713   = wceq 1395   ∈ wcel 2200   ⊆ wss 3197  {csn 3666  {cpr 3667  ⟨cop 3669  BOUNDED wbdc 16161

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 16134  ax-bdan 16136  ax-bdor 16137  ax-bdal 16139  ax-bdeq 16141  ax-bdel 16142  ax-bdsb 16143

This theorem depends on definitions:  df-bi 117  df-3an 1004  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 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-bdc 16162

This theorem is referenced by: (None)