Theorem regexmidlemm 4385

Description: Lemma for regexmid 4388. 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 𝑦 ∈ 𝐴

Distinct variable groups:   𝑦,𝐴   𝜑,𝑥,𝑦

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem regexmidlemm

Step	Hyp	Ref	Expression
1		p0ex 4052	. . . 4 ⊢ {∅} ∈ V
2	1	prid2 3577	. . 3 ⊢ {∅} ∈ {∅, {∅}}
3		eqid 2100	. . . 4 ⊢ {∅} = {∅}
4	3	orci 691	. . 3 ⊢ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑))
5		eqeq1 2106	. . . . 5 ⊢ (𝑥 = {∅} → (𝑥 = {∅} ↔ {∅} = {∅}))
6		eqeq1 2106	. . . . . 6 ⊢ (𝑥 = {∅} → (𝑥 = ∅ ↔ {∅} = ∅))
7	6	anbi1d 456	. . . . 5 ⊢ (𝑥 = {∅} → ((𝑥 = ∅ ∧ 𝜑) ↔ ({∅} = ∅ ∧ 𝜑)))
8	5, 7	orbi12d 748	. . . 4 ⊢ (𝑥 = {∅} → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑))))
9		regexmidlemm.a	. . . 4 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
10	8, 9	elrab2 2796	. . 3 ⊢ ({∅} ∈ 𝐴 ↔ ({∅} ∈ {∅, {∅}} ∧ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑))))
11	2, 4, 10	mpbir2an 894	. 2 ⊢ {∅} ∈ 𝐴
12		elex2 2657	. 2 ⊢ ({∅} ∈ 𝐴 → ∃𝑦 𝑦 ∈ 𝐴)
13	11, 12	ax-mp 7	1 ⊢ ∃𝑦 𝑦 ∈ 𝐴

Syntax hints:   ∧ wa 103   ∨ wo 670   = wceq 1299  ∃wex 1436   ∈ wcel 1448  {crab 2379  ∅c0 3310  {csn 3474  {cpr 3475

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rab 2384  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481

This theorem is referenced by:  regexmid  4388  reg2exmid  4389  reg3exmid  4432  nnregexmid  4472