ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  uniop GIF version

Theorem uniop 4257
Description: The union of an ordered pair. Theorem 65 of [Suppes] p. 39. (Contributed by NM, 17-Aug-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
opthw.1 𝐴 ∈ V
opthw.2 𝐵 ∈ V
Assertion
Ref Expression
uniop 𝐴, 𝐵⟩ = {𝐴, 𝐵}

Proof of Theorem uniop
StepHypRef Expression
1 opthw.1 . . . 4 𝐴 ∈ V
2 opthw.2 . . . 4 𝐵 ∈ V
31, 2dfop 3779 . . 3 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
43unieqi 3821 . 2 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
51snex 4187 . . 3 {𝐴} ∈ V
6 prexg 4213 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
71, 2, 6mp2an 426 . . 3 {𝐴, 𝐵} ∈ V
85, 7unipr 3825 . 2 {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∪ {𝐴, 𝐵})
9 snsspr1 3742 . . 3 {𝐴} ⊆ {𝐴, 𝐵}
10 ssequn1 3307 . . 3 ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵})
119, 10mpbi 145 . 2 ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵}
124, 8, 113eqtri 2202 1 𝐴, 𝐵⟩ = {𝐴, 𝐵}
Colors of variables: wff set class
Syntax hints:   = wceq 1353  wcel 2148  Vcvv 2739  cun 3129  wss 3131  {csn 3594  {cpr 3595  cop 3597   cuni 3811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812
This theorem is referenced by:  uniopel  4258  elvvuni  4692  dmrnssfld  4892
  Copyright terms: Public domain W3C validator