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Theorem regexmidlemm 4284
Description: Lemma for regexmid 4287. 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 3966 . . . 4 {∅} ∈ V
21prid2 3504 . . 3 {∅} ∈ {∅, {∅}}
3 eqid 2056 . . . 4 {∅} = {∅}
43orci 660 . . 3 ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑))
5 eqeq1 2062 . . . . 5 (𝑥 = {∅} → (𝑥 = {∅} ↔ {∅} = {∅}))
6 eqeq1 2062 . . . . . 6 (𝑥 = {∅} → (𝑥 = ∅ ↔ {∅} = ∅))
76anbi1d 446 . . . . 5 (𝑥 = {∅} → ((𝑥 = ∅ ∧ 𝜑) ↔ ({∅} = ∅ ∧ 𝜑)))
85, 7orbi12d 717 . . . 4 (𝑥 = {∅} → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑))))
9 regexmidlemm.a . . . 4 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
108, 9elrab2 2722 . . 3 ({∅} ∈ 𝐴 ↔ ({∅} ∈ {∅, {∅}} ∧ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑))))
112, 4, 10mpbir2an 860 . 2 {∅} ∈ 𝐴
12 elex2 2587 . 2 ({∅} ∈ 𝐴 → ∃𝑦 𝑦𝐴)
1311, 12ax-mp 7 1 𝑦 𝑦𝐴
Colors of variables: wff set class
Syntax hints:  wa 101  wo 639   = wceq 1259  wex 1397  wcel 1409  {crab 2327  c0 3251  {csn 3402  {cpr 3403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-nul 3910  ax-pow 3954
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rab 2332  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409
This theorem is referenced by:  regexmid  4287  reg2exmid  4288  reg3exmid  4331  nnregexmid  4369
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