Theorem bdop 14597

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 14596	. . . 4 ⊢ BOUNDED 𝑧 = {𝑥}
2		bdcpr 14593	. . . . . . 7 ⊢ BOUNDED {𝑥, 𝑦}
3	2	bdss 14586	. . . . . 6 ⊢ BOUNDED 𝑧 ⊆ {𝑥, 𝑦}
4		ax-bdel 14543	. . . . . . . 8 ⊢ BOUNDED 𝑥 ∈ 𝑧
5		ax-bdel 14543	. . . . . . . 8 ⊢ BOUNDED 𝑦 ∈ 𝑧
6	4, 5	ax-bdan 14537	. . . . . . 7 ⊢ BOUNDED (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑧)
7		vex 2740	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
8	7	prid1 3698	. . . . . . . . . 10 ⊢ 𝑥 ∈ {𝑥, 𝑦}
9		ssel 3149	. . . . . . . . . 10 ⊢ ({𝑥, 𝑦} ⊆ 𝑧 → (𝑥 ∈ {𝑥, 𝑦} → 𝑥 ∈ 𝑧))
10	8, 9	mpi 15	. . . . . . . . 9 ⊢ ({𝑥, 𝑦} ⊆ 𝑧 → 𝑥 ∈ 𝑧)
11		vex 2740	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
12	11	prid2 3699	. . . . . . . . . 10 ⊢ 𝑦 ∈ {𝑥, 𝑦}
13		ssel 3149	. . . . . . . . . 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 3750	. . . . . . . 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 14547	. . . . . 6 ⊢ BOUNDED {𝑥, 𝑦} ⊆ 𝑧
19	3, 18	ax-bdan 14537	. . . . 5 ⊢ BOUNDED (𝑧 ⊆ {𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑧)
20		eqss 3170	. . . . 5 ⊢ (𝑧 = {𝑥, 𝑦} ↔ (𝑧 ⊆ {𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑧))
21	19, 20	bd0r 14547	. . . 4 ⊢ BOUNDED 𝑧 = {𝑥, 𝑦}
22	1, 21	ax-bdor 14538	. . 3 ⊢ BOUNDED (𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦})
23		vex 2740	. . . 4 ⊢ 𝑧 ∈ V
24	23, 7, 11	elop 4231	. . 3 ⊢ (𝑧 ∈ ⟨𝑥, 𝑦⟩ ↔ (𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦}))
25	22, 24	bd0r 14547	. 2 ⊢ BOUNDED 𝑧 ∈ ⟨𝑥, 𝑦⟩
26	25	bdelir 14569	1 ⊢ BOUNDED ⟨𝑥, 𝑦⟩

Syntax hints:   ∧ wa 104   ∨ wo 708   = wceq 1353   ∈ wcel 2148   ⊆ wss 3129  {csn 3592  {cpr 3593  ⟨cop 3595  BOUNDED wbdc 14562

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 14535  ax-bdan 14537  ax-bdor 14538  ax-bdal 14540  ax-bdeq 14542  ax-bdel 14543  ax-bdsb 14544

This theorem depends on definitions:  df-bi 117  df-3an 980  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 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599  df-op 3601  df-bdc 14563

This theorem is referenced by: (None)