Theorem opexgOLD 3965

Description: An ordered pair of sets is a set. This is a special case of opexg 3964 and new proofs should use opexg 3964 instead. (Contributed by Jim Kingdon, 19-Sep-2018.) (New usage is discouraged.) (Proof modification is discouraged.) TODO: replace its uses by uses of opexg 3964 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 3547	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
2		snexgOLD 3935	. . . . 5 ⊢ (𝐴 ∈ V → {𝐴} ∈ V)
3	2	adantr 261	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴} ∈ V)
4		prexgOLD 3946	. . . 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 3946	. . 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 2114	1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ V)

Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1393  Vcvv 2557  {csn 3375  {cpr 3376  ⟨cop 3378

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384

This theorem is referenced by:  otth2  3978  opeliunxp  4395  opbrop  4419  relsnop  4444  op2ndb  4804  opswapg  4807  elxp4  4808  elxp5  4809  fvsn  5358  resfunexg  5382  fliftel  5433