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

Theorem regexmidlemm 4549
Description: Lemma for regexmid 4552. 𝐴 is inhabited. (Contributed by Jim Kingdon, 3-Sep-2019.)
Hypothesis
Ref Expression
regexmidlemm.a 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
Assertion
Ref Expression
regexmidlemm 𝑦 𝑦𝐴
Distinct variable groups:   𝑦,𝐴   𝜑,𝑥,𝑦
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem regexmidlemm
StepHypRef Expression
1 p0ex 4206 . . . 4 {∅} ∈ V
21prid2 3714 . . 3 {∅} ∈ {∅, {∅}}
3 eqid 2189 . . . 4 {∅} = {∅}
43orci 732 . . 3 ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑))
5 eqeq1 2196 . . . . 5 (𝑥 = {∅} → (𝑥 = {∅} ↔ {∅} = {∅}))
6 eqeq1 2196 . . . . . 6 (𝑥 = {∅} → (𝑥 = ∅ ↔ {∅} = ∅))
76anbi1d 465 . . . . 5 (𝑥 = {∅} → ((𝑥 = ∅ ∧ 𝜑) ↔ ({∅} = ∅ ∧ 𝜑)))
85, 7orbi12d 794 . . . 4 (𝑥 = {∅} → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑))))
9 regexmidlemm.a . . . 4 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
108, 9elrab2 2911 . . 3 ({∅} ∈ 𝐴 ↔ ({∅} ∈ {∅, {∅}} ∧ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑))))
112, 4, 10mpbir2an 944 . 2 {∅} ∈ 𝐴
12 elex2 2768 . 2 ({∅} ∈ 𝐴 → ∃𝑦 𝑦𝐴)
1311, 12ax-mp 5 1 𝑦 𝑦𝐴
Colors of variables: wff set class
Syntax hints:  wa 104  wo 709   = wceq 1364  wex 1503  wcel 2160  {crab 2472  c0 3437  {csn 3607  {cpr 3608
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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614
This theorem is referenced by:  regexmid  4552  reg2exmid  4553  reg3exmid  4597  nnregexmid  4638
  Copyright terms: Public domain W3C validator