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

Theorem nfop 3633
Description: Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.)
Hypotheses
Ref Expression
nfop.1 𝑥𝐴
nfop.2 𝑥𝐵
Assertion
Ref Expression
nfop 𝑥𝐴, 𝐵

Proof of Theorem nfop
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-op 3450 . 2 𝐴, 𝐵⟩ = {𝑦 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})}
2 nfop.1 . . . . 5 𝑥𝐴
32nfel1 2239 . . . 4 𝑥 𝐴 ∈ V
4 nfop.2 . . . . 5 𝑥𝐵
54nfel1 2239 . . . 4 𝑥 𝐵 ∈ V
62nfsn 3497 . . . . . 6 𝑥{𝐴}
72, 4nfpr 3487 . . . . . 6 𝑥{𝐴, 𝐵}
86, 7nfpr 3487 . . . . 5 𝑥{{𝐴}, {𝐴, 𝐵}}
98nfcri 2222 . . . 4 𝑥 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}}
103, 5, 9nf3an 1503 . . 3 𝑥(𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})
1110nfab 2233 . 2 𝑥{𝑦 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑦 ∈ {{𝐴}, {𝐴, 𝐵}})}
121, 11nfcxfr 2225 1 𝑥𝐴, 𝐵
Colors of variables: wff set class
Syntax hints:  w3a 924  wcel 1438  {cab 2074  wnfc 2215  Vcvv 2619  {csn 3441  {cpr 3442  cop 3444
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448  df-op 3450
This theorem is referenced by:  nfopd  3634  moop2  4069  fliftfuns  5559  dfmpt2  5970  qliftfuns  6356  xpf1o  6540  caucvgprprlemaddq  7246  nfiseq  9833
  Copyright terms: Public domain W3C validator