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Theorem bdop 13000
Description: The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.)
Assertion
Ref Expression
bdop BOUNDED𝑥, 𝑦

Proof of Theorem bdop
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 bdvsn 12999 . . . 4 BOUNDED 𝑧 = {𝑥}
2 bdcpr 12996 . . . . . . 7 BOUNDED {𝑥, 𝑦}
32bdss 12989 . . . . . 6 BOUNDED 𝑧 ⊆ {𝑥, 𝑦}
4 ax-bdel 12946 . . . . . . . 8 BOUNDED 𝑥𝑧
5 ax-bdel 12946 . . . . . . . 8 BOUNDED 𝑦𝑧
64, 5ax-bdan 12940 . . . . . . 7 BOUNDED (𝑥𝑧𝑦𝑧)
7 vex 2663 . . . . . . . . . . 11 𝑥 ∈ V
87prid1 3599 . . . . . . . . . 10 𝑥 ∈ {𝑥, 𝑦}
9 ssel 3061 . . . . . . . . . 10 ({𝑥, 𝑦} ⊆ 𝑧 → (𝑥 ∈ {𝑥, 𝑦} → 𝑥𝑧))
108, 9mpi 15 . . . . . . . . 9 ({𝑥, 𝑦} ⊆ 𝑧𝑥𝑧)
11 vex 2663 . . . . . . . . . . 11 𝑦 ∈ V
1211prid2 3600 . . . . . . . . . 10 𝑦 ∈ {𝑥, 𝑦}
13 ssel 3061 . . . . . . . . . 10 ({𝑥, 𝑦} ⊆ 𝑧 → (𝑦 ∈ {𝑥, 𝑦} → 𝑦𝑧))
1412, 13mpi 15 . . . . . . . . 9 ({𝑥, 𝑦} ⊆ 𝑧𝑦𝑧)
1510, 14jca 304 . . . . . . . 8 ({𝑥, 𝑦} ⊆ 𝑧 → (𝑥𝑧𝑦𝑧))
16 prssi 3648 . . . . . . . 8 ((𝑥𝑧𝑦𝑧) → {𝑥, 𝑦} ⊆ 𝑧)
1715, 16impbii 125 . . . . . . 7 ({𝑥, 𝑦} ⊆ 𝑧 ↔ (𝑥𝑧𝑦𝑧))
186, 17bd0r 12950 . . . . . 6 BOUNDED {𝑥, 𝑦} ⊆ 𝑧
193, 18ax-bdan 12940 . . . . 5 BOUNDED (𝑧 ⊆ {𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑧)
20 eqss 3082 . . . . 5 (𝑧 = {𝑥, 𝑦} ↔ (𝑧 ⊆ {𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑧))
2119, 20bd0r 12950 . . . 4 BOUNDED 𝑧 = {𝑥, 𝑦}
221, 21ax-bdor 12941 . . 3 BOUNDED (𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦})
23 vex 2663 . . . 4 𝑧 ∈ V
2423, 7, 11elop 4123 . . 3 (𝑧 ∈ ⟨𝑥, 𝑦⟩ ↔ (𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦}))
2522, 24bd0r 12950 . 2 BOUNDED 𝑧 ∈ ⟨𝑥, 𝑦
2625bdelir 12972 1 BOUNDED𝑥, 𝑦
Colors of variables: wff set class
Syntax hints:  wa 103  wo 682   = wceq 1316  wcel 1465  wss 3041  {csn 3497  {cpr 3498  cop 3500  BOUNDED wbdc 12965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-bd0 12938  ax-bdan 12940  ax-bdor 12941  ax-bdal 12943  ax-bdeq 12945  ax-bdel 12946  ax-bdsb 12947
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-pr 3504  df-op 3506  df-bdc 12966
This theorem is referenced by: (None)
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