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Theorem bdop 10824
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 10823 . . . 4 BOUNDED 𝑧 = {𝑥}
2 bdcpr 10820 . . . . . . 7 BOUNDED {𝑥, 𝑦}
32bdss 10813 . . . . . 6 BOUNDED 𝑧 ⊆ {𝑥, 𝑦}
4 ax-bdel 10770 . . . . . . . 8 BOUNDED 𝑥𝑧
5 ax-bdel 10770 . . . . . . . 8 BOUNDED 𝑦𝑧
64, 5ax-bdan 10764 . . . . . . 7 BOUNDED (𝑥𝑧𝑦𝑧)
7 vex 2605 . . . . . . . . . . 11 𝑥 ∈ V
87prid1 3506 . . . . . . . . . 10 𝑥 ∈ {𝑥, 𝑦}
9 ssel 2994 . . . . . . . . . 10 ({𝑥, 𝑦} ⊆ 𝑧 → (𝑥 ∈ {𝑥, 𝑦} → 𝑥𝑧))
108, 9mpi 15 . . . . . . . . 9 ({𝑥, 𝑦} ⊆ 𝑧𝑥𝑧)
11 vex 2605 . . . . . . . . . . 11 𝑦 ∈ V
1211prid2 3507 . . . . . . . . . 10 𝑦 ∈ {𝑥, 𝑦}
13 ssel 2994 . . . . . . . . . 10 ({𝑥, 𝑦} ⊆ 𝑧 → (𝑦 ∈ {𝑥, 𝑦} → 𝑦𝑧))
1412, 13mpi 15 . . . . . . . . 9 ({𝑥, 𝑦} ⊆ 𝑧𝑦𝑧)
1510, 14jca 300 . . . . . . . 8 ({𝑥, 𝑦} ⊆ 𝑧 → (𝑥𝑧𝑦𝑧))
16 prssi 3551 . . . . . . . 8 ((𝑥𝑧𝑦𝑧) → {𝑥, 𝑦} ⊆ 𝑧)
1715, 16impbii 124 . . . . . . 7 ({𝑥, 𝑦} ⊆ 𝑧 ↔ (𝑥𝑧𝑦𝑧))
186, 17bd0r 10774 . . . . . 6 BOUNDED {𝑥, 𝑦} ⊆ 𝑧
193, 18ax-bdan 10764 . . . . 5 BOUNDED (𝑧 ⊆ {𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑧)
20 eqss 3015 . . . . 5 (𝑧 = {𝑥, 𝑦} ↔ (𝑧 ⊆ {𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑧))
2119, 20bd0r 10774 . . . 4 BOUNDED 𝑧 = {𝑥, 𝑦}
221, 21ax-bdor 10765 . . 3 BOUNDED (𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦})
23 vex 2605 . . . 4 𝑧 ∈ V
2423, 7, 11elop 3994 . . 3 (𝑧 ∈ ⟨𝑥, 𝑦⟩ ↔ (𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦}))
2522, 24bd0r 10774 . 2 BOUNDED 𝑧 ∈ ⟨𝑥, 𝑦
2625bdelir 10796 1 BOUNDED𝑥, 𝑦
Colors of variables: wff set class
Syntax hints:  wa 102  wo 662   = wceq 1285  wcel 1434  wss 2974  {csn 3406  {cpr 3407  cop 3409  BOUNDED wbdc 10789
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-bd0 10762  ax-bdan 10764  ax-bdor 10765  ax-bdal 10767  ax-bdeq 10769  ax-bdel 10770  ax-bdsb 10771
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3412  df-pr 3413  df-op 3415  df-bdc 10790
This theorem is referenced by: (None)
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