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Theorem bdop 12907
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 12906 . . . 4 BOUNDED 𝑧 = {𝑥}
2 bdcpr 12903 . . . . . . 7 BOUNDED {𝑥, 𝑦}
32bdss 12896 . . . . . 6 BOUNDED 𝑧 ⊆ {𝑥, 𝑦}
4 ax-bdel 12853 . . . . . . . 8 BOUNDED 𝑥𝑧
5 ax-bdel 12853 . . . . . . . 8 BOUNDED 𝑦𝑧
64, 5ax-bdan 12847 . . . . . . 7 BOUNDED (𝑥𝑧𝑦𝑧)
7 vex 2661 . . . . . . . . . . 11 𝑥 ∈ V
87prid1 3597 . . . . . . . . . 10 𝑥 ∈ {𝑥, 𝑦}
9 ssel 3059 . . . . . . . . . 10 ({𝑥, 𝑦} ⊆ 𝑧 → (𝑥 ∈ {𝑥, 𝑦} → 𝑥𝑧))
108, 9mpi 15 . . . . . . . . 9 ({𝑥, 𝑦} ⊆ 𝑧𝑥𝑧)
11 vex 2661 . . . . . . . . . . 11 𝑦 ∈ V
1211prid2 3598 . . . . . . . . . 10 𝑦 ∈ {𝑥, 𝑦}
13 ssel 3059 . . . . . . . . . 10 ({𝑥, 𝑦} ⊆ 𝑧 → (𝑦 ∈ {𝑥, 𝑦} → 𝑦𝑧))
1412, 13mpi 15 . . . . . . . . 9 ({𝑥, 𝑦} ⊆ 𝑧𝑦𝑧)
1510, 14jca 302 . . . . . . . 8 ({𝑥, 𝑦} ⊆ 𝑧 → (𝑥𝑧𝑦𝑧))
16 prssi 3646 . . . . . . . 8 ((𝑥𝑧𝑦𝑧) → {𝑥, 𝑦} ⊆ 𝑧)
1715, 16impbii 125 . . . . . . 7 ({𝑥, 𝑦} ⊆ 𝑧 ↔ (𝑥𝑧𝑦𝑧))
186, 17bd0r 12857 . . . . . 6 BOUNDED {𝑥, 𝑦} ⊆ 𝑧
193, 18ax-bdan 12847 . . . . 5 BOUNDED (𝑧 ⊆ {𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑧)
20 eqss 3080 . . . . 5 (𝑧 = {𝑥, 𝑦} ↔ (𝑧 ⊆ {𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑧))
2119, 20bd0r 12857 . . . 4 BOUNDED 𝑧 = {𝑥, 𝑦}
221, 21ax-bdor 12848 . . 3 BOUNDED (𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦})
23 vex 2661 . . . 4 𝑧 ∈ V
2423, 7, 11elop 4121 . . 3 (𝑧 ∈ ⟨𝑥, 𝑦⟩ ↔ (𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦}))
2522, 24bd0r 12857 . 2 BOUNDED 𝑧 ∈ ⟨𝑥, 𝑦
2625bdelir 12879 1 BOUNDED𝑥, 𝑦
Colors of variables: wff set class
Syntax hints:  wa 103  wo 680   = wceq 1314  wcel 1463  wss 3039  {csn 3495  {cpr 3496  cop 3498  BOUNDED wbdc 12872
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-bd0 12845  ax-bdan 12847  ax-bdor 12848  ax-bdal 12850  ax-bdeq 12852  ax-bdel 12853  ax-bdsb 12854
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-sn 3501  df-pr 3502  df-op 3504  df-bdc 12873
This theorem is referenced by: (None)
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