Theorem opexgOLD 3939

Description: An ordered pair of sets is a set. This is a special case of opexg 3938 and new proofs should use opexg 3938 instead. (Contributed by Jim Kingdon, 19-Sep-2018.) (New usage is discouraged.) TODO: remove in favor of opexg 3938.

Assertion

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

Proof of Theorem opexgOLD

Step	Hyp	Ref	Expression
1		dfopg 3521	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → ⟨A, B⟩ = {{A}, {A, B}})
2		snexgOLD 3909	. . . . 5 ⊢ (A ∈ V → {A} ∈ V)
3	2	adantr 261	. . . 4 ⊢ ((A ∈ V ∧ B ∈ V) → {A} ∈ V)
4		prexgOLD 3920	. . . 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 3920	. . 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 2096	1 ⊢ ((A ∈ V ∧ B ∈ V) → ⟨A, B⟩ ∈ V)

Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1374  Vcvv 2535  {csn 3350  {cpr 3351  ⟨cop 3353

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359

This theorem is referenced by:  otth2  3952  opeliunxp  4322  opbrop  4346  relsnop  4371  op2ndb  4731  opswapg  4734  elxp4  4735  elxp5  4736  fvsn  5283  resfunexg  5307  fliftel  5358