Theorem opexgOLD 3956

Description: An ordered pair of sets is a set. This is a special case of opexg 3955 and new proofs should use opexg 3955 instead. (Contributed by Jim Kingdon, 19-Sep-2018.) (New usage is discouraged.) (Proof modification is discouraged.) TODO: replace its uses by uses of opexg 3955 and then remove it.

Assertion

Ref	Expression
opexgOLD	⊢ ((A ∈ V ∧ B ∈ V) → ⟨A, B⟩ ∈ V)

Proof of Theorem opexgOLD

Step	Hyp	Ref	Expression
1		dfopg 3538	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → ⟨A, B⟩ = {{A}, {A, B}})
2		snexgOLD 3926	. . . . 5 ⊢ (A ∈ V → {A} ∈ V)
3	2	adantr 261	. . . 4 ⊢ ((A ∈ V ∧ B ∈ V) → {A} ∈ V)
4		prexgOLD 3937	. . . 4 ⊢ ((A ∈ V ∧ B ∈ V) → {A, B} ∈ V)
5	3, 4	jca 290	. . 3 ⊢ ((A ∈ V ∧ B ∈ V) → ({A} ∈ V ∧ {A, B} ∈ V))
6		prexgOLD 3937	. . 3 ⊢ (({A} ∈ V ∧ {A, B} ∈ V) → {{A}, {A, B}} ∈ V)
7	5, 6	syl 14	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → {{A}, {A, B}} ∈ V)
8	1, 7	eqeltrd 2111	1 ⊢ ((A ∈ V ∧ B ∈ V) → ⟨A, B⟩ ∈ V)

Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1390  Vcvv 2551  {csn 3367  {cpr 3368  ⟨cop 3370

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376

This theorem is referenced by:  otth2  3969  opeliunxp  4338  opbrop  4362  relsnop  4387  op2ndb  4747  opswapg  4750  elxp4  4751  elxp5  4752  fvsn  5301  resfunexg  5325  fliftel  5376