Theorem bdop 10666

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 10665	. . . 4 ⊢ BOUNDED 𝑧 = {𝑥}
2		bdcpr 10662	. . . . . . 7 ⊢ BOUNDED {𝑥, 𝑦}
3	2	bdss 10655	. . . . . 6 ⊢ BOUNDED 𝑧 ⊆ {𝑥, 𝑦}
4		ax-bdel 10612	. . . . . . . 8 ⊢ BOUNDED 𝑥 ∈ 𝑧
5		ax-bdel 10612	. . . . . . . 8 ⊢ BOUNDED 𝑦 ∈ 𝑧
6	4, 5	ax-bdan 10606	. . . . . . 7 ⊢ BOUNDED (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑧)
7		vex 2604	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
8	7	prid1 3498	. . . . . . . . . 10 ⊢ 𝑥 ∈ {𝑥, 𝑦}
9		ssel 2993	. . . . . . . . . 10 ⊢ ({𝑥, 𝑦} ⊆ 𝑧 → (𝑥 ∈ {𝑥, 𝑦} → 𝑥 ∈ 𝑧))
10	8, 9	mpi 15	. . . . . . . . 9 ⊢ ({𝑥, 𝑦} ⊆ 𝑧 → 𝑥 ∈ 𝑧)
11		vex 2604	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
12	11	prid2 3499	. . . . . . . . . 10 ⊢ 𝑦 ∈ {𝑥, 𝑦}
13		ssel 2993	. . . . . . . . . 10 ⊢ ({𝑥, 𝑦} ⊆ 𝑧 → (𝑦 ∈ {𝑥, 𝑦} → 𝑦 ∈ 𝑧))
14	12, 13	mpi 15	. . . . . . . . 9 ⊢ ({𝑥, 𝑦} ⊆ 𝑧 → 𝑦 ∈ 𝑧)
15	10, 14	jca 300	. . . . . . . 8 ⊢ ({𝑥, 𝑦} ⊆ 𝑧 → (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑧))
16		prssi 3543	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑧) → {𝑥, 𝑦} ⊆ 𝑧)
17	15, 16	impbii 124	. . . . . . 7 ⊢ ({𝑥, 𝑦} ⊆ 𝑧 ↔ (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑧))
18	6, 17	bd0r 10616	. . . . . 6 ⊢ BOUNDED {𝑥, 𝑦} ⊆ 𝑧
19	3, 18	ax-bdan 10606	. . . . 5 ⊢ BOUNDED (𝑧 ⊆ {𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑧)
20		eqss 3014	. . . . 5 ⊢ (𝑧 = {𝑥, 𝑦} ↔ (𝑧 ⊆ {𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑧))
21	19, 20	bd0r 10616	. . . 4 ⊢ BOUNDED 𝑧 = {𝑥, 𝑦}
22	1, 21	ax-bdor 10607	. . 3 ⊢ BOUNDED (𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦})
23		vex 2604	. . . 4 ⊢ 𝑧 ∈ V
24	23, 7, 11	elop 3986	. . 3 ⊢ (𝑧 ∈ ⟨𝑥, 𝑦⟩ ↔ (𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦}))
25	22, 24	bd0r 10616	. 2 ⊢ BOUNDED 𝑧 ∈ ⟨𝑥, 𝑦⟩
26	25	bdelir 10638	1 ⊢ BOUNDED ⟨𝑥, 𝑦⟩

Syntax hints:   ∧ wa 102   ∨ wo 661   = wceq 1284   ∈ wcel 1433   ⊆ wss 2973  {csn 3398  {cpr 3399  ⟨cop 3401  BOUNDED wbdc 10631

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-bd0 10604  ax-bdan 10606  ax-bdor 10607  ax-bdal 10609  ax-bdeq 10611  ax-bdel 10612  ax-bdsb 10613

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-pr 3405  df-op 3407  df-bdc 10632

This theorem is referenced by: (None)