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Theorem opexgOLD 3993
Description: An ordered pair of sets is a set. This is a special case of opexg 3992 and new proofs should use opexg 3992 instead. (Contributed by Jim Kingdon, 19-Sep-2018.) (New usage is discouraged.) (Proof modification is discouraged.) TODO: replace its uses by uses of opexg 3992 and then remove it.
Assertion
Ref Expression
opexgOLD ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ V)

Proof of Theorem opexgOLD
StepHypRef Expression
1 dfopg 3575 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
2 snexgOLD 3963 . . . . 5 (𝐴 ∈ V → {𝐴} ∈ V)
32adantr 265 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴} ∈ V)
4 prexgOLD 3974 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
53, 4jca 294 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴} ∈ V ∧ {𝐴, 𝐵} ∈ V))
6 prexgOLD 3974 . . 3 (({𝐴} ∈ V ∧ {𝐴, 𝐵} ∈ V) → {{𝐴}, {𝐴, 𝐵}} ∈ V)
75, 6syl 14 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {{𝐴}, {𝐴, 𝐵}} ∈ V)
81, 7eqeltrd 2130 1 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wcel 1409  Vcvv 2574  {csn 3403  {cpr 3404  cop 3406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412
This theorem is referenced by:  otth2  4006  opeliunxp  4423  opbrop  4447  relsnop  4472  op2ndb  4832  opswapg  4835  elxp4  4836  elxp5  4837  fvsn  5386  resfunexg  5410  fliftel  5461
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