Theorem opm 3962

Description: An ordered pair is inhabited iff the arguments are sets. (Contributed by Jim Kingdon, 21-Sep-2018.)

Assertion

Ref	Expression
opm	⊢ (∃x x ∈ ⟨A, B⟩ ↔ (A ∈ V ∧ B ∈ V))

Distinct variable groups:   x,A   x,B

Proof of Theorem opm

Step	Hyp	Ref	Expression
1		df-op 3376	. . . . 5 ⊢ ⟨A, B⟩ = {x ∣ (A ∈ V ∧ B ∈ V ∧ x ∈ {{A}, {A, B}})}
2	1	eleq2i 2101	. . . 4 ⊢ (x ∈ ⟨A, B⟩ ↔ x ∈ {x ∣ (A ∈ V ∧ B ∈ V ∧ x ∈ {{A}, {A, B}})})
3	2	exbii 1493	. . 3 ⊢ (∃x x ∈ ⟨A, B⟩ ↔ ∃x x ∈ {x ∣ (A ∈ V ∧ B ∈ V ∧ x ∈ {{A}, {A, B}})})
4		abid 2025	. . . 4 ⊢ (x ∈ {x ∣ (A ∈ V ∧ B ∈ V ∧ x ∈ {{A}, {A, B}})} ↔ (A ∈ V ∧ B ∈ V ∧ x ∈ {{A}, {A, B}}))
5	4	exbii 1493	. . 3 ⊢ (∃x x ∈ {x ∣ (A ∈ V ∧ B ∈ V ∧ x ∈ {{A}, {A, B}})} ↔ ∃x(A ∈ V ∧ B ∈ V ∧ x ∈ {{A}, {A, B}}))
6	3, 5	bitri 173	. 2 ⊢ (∃x x ∈ ⟨A, B⟩ ↔ ∃x(A ∈ V ∧ B ∈ V ∧ x ∈ {{A}, {A, B}}))
7		19.42v 1783	. . 3 ⊢ (∃x((A ∈ V ∧ B ∈ V) ∧ x ∈ {{A}, {A, B}}) ↔ ((A ∈ V ∧ B ∈ V) ∧ ∃x x ∈ {{A}, {A, B}}))
8		df-3an 886	. . . 4 ⊢ ((A ∈ V ∧ B ∈ V ∧ x ∈ {{A}, {A, B}}) ↔ ((A ∈ V ∧ B ∈ V) ∧ x ∈ {{A}, {A, B}}))
9	8	exbii 1493	. . 3 ⊢ (∃x(A ∈ V ∧ B ∈ V ∧ x ∈ {{A}, {A, B}}) ↔ ∃x((A ∈ V ∧ B ∈ V) ∧ x ∈ {{A}, {A, B}}))
10		df-3an 886	. . 3 ⊢ ((A ∈ V ∧ B ∈ V ∧ ∃x x ∈ {{A}, {A, B}}) ↔ ((A ∈ V ∧ B ∈ V) ∧ ∃x x ∈ {{A}, {A, B}}))
11	7, 9, 10	3bitr4ri 202	. 2 ⊢ ((A ∈ V ∧ B ∈ V ∧ ∃x x ∈ {{A}, {A, B}}) ↔ ∃x(A ∈ V ∧ B ∈ V ∧ x ∈ {{A}, {A, B}}))
12		3simpa 900	. . 3 ⊢ ((A ∈ V ∧ B ∈ V ∧ ∃x x ∈ {{A}, {A, B}}) → (A ∈ V ∧ B ∈ V))
13		id 19	. . . 4 ⊢ ((A ∈ V ∧ B ∈ V) → (A ∈ V ∧ B ∈ V))
14		snexgOLD 3926	. . . . . 6 ⊢ (A ∈ V → {A} ∈ V)
15	14	adantr 261	. . . . 5 ⊢ ((A ∈ V ∧ B ∈ V) → {A} ∈ V)
16		prmg 3480	. . . . 5 ⊢ ({A} ∈ V → ∃x x ∈ {{A}, {A, B}})
17	15, 16	syl 14	. . . 4 ⊢ ((A ∈ V ∧ B ∈ V) → ∃x x ∈ {{A}, {A, B}})
18	13, 17, 10	sylanbrc 394	. . 3 ⊢ ((A ∈ V ∧ B ∈ V) → (A ∈ V ∧ B ∈ V ∧ ∃x x ∈ {{A}, {A, B}}))
19	12, 18	impbii 117	. 2 ⊢ ((A ∈ V ∧ B ∈ V ∧ ∃x x ∈ {{A}, {A, B}}) ↔ (A ∈ V ∧ B ∈ V))
20	6, 11, 19	3bitr2i 197	1 ⊢ (∃x x ∈ ⟨A, B⟩ ↔ (A ∈ V ∧ B ∈ V))

Syntax hints:   ∧ wa 97   ↔ wb 98   ∧ w3a 884  ∃wex 1378   ∈ wcel 1390  {cab 2023  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-bnd 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

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:  opnzi  3963  opeqex  3977