Theorem bdop 15980

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 15979	. . . 4 ⊢ BOUNDED 𝑧 = {𝑥}
2		bdcpr 15976	. . . . . . 7 ⊢ BOUNDED {𝑥, 𝑦}
3	2	bdss 15969	. . . . . 6 ⊢ BOUNDED 𝑧 ⊆ {𝑥, 𝑦}
4		ax-bdel 15926	. . . . . . . 8 ⊢ BOUNDED 𝑥 ∈ 𝑧
5		ax-bdel 15926	. . . . . . . 8 ⊢ BOUNDED 𝑦 ∈ 𝑧
6	4, 5	ax-bdan 15920	. . . . . . 7 ⊢ BOUNDED (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑧)
7		vex 2776	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
8	7	prid1 3744	. . . . . . . . . 10 ⊢ 𝑥 ∈ {𝑥, 𝑦}
9		ssel 3191	. . . . . . . . . 10 ⊢ ({𝑥, 𝑦} ⊆ 𝑧 → (𝑥 ∈ {𝑥, 𝑦} → 𝑥 ∈ 𝑧))
10	8, 9	mpi 15	. . . . . . . . 9 ⊢ ({𝑥, 𝑦} ⊆ 𝑧 → 𝑥 ∈ 𝑧)
11		vex 2776	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
12	11	prid2 3745	. . . . . . . . . 10 ⊢ 𝑦 ∈ {𝑥, 𝑦}
13		ssel 3191	. . . . . . . . . 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 3797	. . . . . . . 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 15930	. . . . . 6 ⊢ BOUNDED {𝑥, 𝑦} ⊆ 𝑧
19	3, 18	ax-bdan 15920	. . . . 5 ⊢ BOUNDED (𝑧 ⊆ {𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑧)
20		eqss 3212	. . . . 5 ⊢ (𝑧 = {𝑥, 𝑦} ↔ (𝑧 ⊆ {𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑧))
21	19, 20	bd0r 15930	. . . 4 ⊢ BOUNDED 𝑧 = {𝑥, 𝑦}
22	1, 21	ax-bdor 15921	. . 3 ⊢ BOUNDED (𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦})
23		vex 2776	. . . 4 ⊢ 𝑧 ∈ V
24	23, 7, 11	elop 4288	. . 3 ⊢ (𝑧 ∈ ⟨𝑥, 𝑦⟩ ↔ (𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦}))
25	22, 24	bd0r 15930	. 2 ⊢ BOUNDED 𝑧 ∈ ⟨𝑥, 𝑦⟩
26	25	bdelir 15952	1 ⊢ BOUNDED ⟨𝑥, 𝑦⟩

Syntax hints:   ∧ wa 104   ∨ wo 710   = wceq 1373   ∈ wcel 2177   ⊆ wss 3170  {csn 3638  {cpr 3639  ⟨cop 3641  BOUNDED wbdc 15945

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188  ax-bd0 15918  ax-bdan 15920  ax-bdor 15921  ax-bdal 15923  ax-bdeq 15925  ax-bdel 15926  ax-bdsb 15927

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-sn 3644  df-pr 3645  df-op 3647  df-bdc 15946

This theorem is referenced by: (None)