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Theorem bdop 15085
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 15084 . . . 4 BOUNDED 𝑧 = {𝑥}
2 bdcpr 15081 . . . . . . 7 BOUNDED {𝑥, 𝑦}
32bdss 15074 . . . . . 6 BOUNDED 𝑧 ⊆ {𝑥, 𝑦}
4 ax-bdel 15031 . . . . . . . 8 BOUNDED 𝑥𝑧
5 ax-bdel 15031 . . . . . . . 8 BOUNDED 𝑦𝑧
64, 5ax-bdan 15025 . . . . . . 7 BOUNDED (𝑥𝑧𝑦𝑧)
7 vex 2755 . . . . . . . . . . 11 𝑥 ∈ V
87prid1 3713 . . . . . . . . . 10 𝑥 ∈ {𝑥, 𝑦}
9 ssel 3164 . . . . . . . . . 10 ({𝑥, 𝑦} ⊆ 𝑧 → (𝑥 ∈ {𝑥, 𝑦} → 𝑥𝑧))
108, 9mpi 15 . . . . . . . . 9 ({𝑥, 𝑦} ⊆ 𝑧𝑥𝑧)
11 vex 2755 . . . . . . . . . . 11 𝑦 ∈ V
1211prid2 3714 . . . . . . . . . 10 𝑦 ∈ {𝑥, 𝑦}
13 ssel 3164 . . . . . . . . . 10 ({𝑥, 𝑦} ⊆ 𝑧 → (𝑦 ∈ {𝑥, 𝑦} → 𝑦𝑧))
1412, 13mpi 15 . . . . . . . . 9 ({𝑥, 𝑦} ⊆ 𝑧𝑦𝑧)
1510, 14jca 306 . . . . . . . 8 ({𝑥, 𝑦} ⊆ 𝑧 → (𝑥𝑧𝑦𝑧))
16 prssi 3765 . . . . . . . 8 ((𝑥𝑧𝑦𝑧) → {𝑥, 𝑦} ⊆ 𝑧)
1715, 16impbii 126 . . . . . . 7 ({𝑥, 𝑦} ⊆ 𝑧 ↔ (𝑥𝑧𝑦𝑧))
186, 17bd0r 15035 . . . . . 6 BOUNDED {𝑥, 𝑦} ⊆ 𝑧
193, 18ax-bdan 15025 . . . . 5 BOUNDED (𝑧 ⊆ {𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑧)
20 eqss 3185 . . . . 5 (𝑧 = {𝑥, 𝑦} ↔ (𝑧 ⊆ {𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑧))
2119, 20bd0r 15035 . . . 4 BOUNDED 𝑧 = {𝑥, 𝑦}
221, 21ax-bdor 15026 . . 3 BOUNDED (𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦})
23 vex 2755 . . . 4 𝑧 ∈ V
2423, 7, 11elop 4249 . . 3 (𝑧 ∈ ⟨𝑥, 𝑦⟩ ↔ (𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦}))
2522, 24bd0r 15035 . 2 BOUNDED 𝑧 ∈ ⟨𝑥, 𝑦
2625bdelir 15057 1 BOUNDED𝑥, 𝑦
Colors of variables: wff set class
Syntax hints:  wa 104  wo 709   = wceq 1364  wcel 2160  wss 3144  {csn 3607  {cpr 3608  cop 3610  BOUNDED wbdc 15050
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-bd0 15023  ax-bdan 15025  ax-bdor 15026  ax-bdal 15028  ax-bdeq 15030  ax-bdel 15031  ax-bdsb 15032
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-op 3616  df-bdc 15051
This theorem is referenced by: (None)
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