Theorem regexmidlemm 4593

Description: Lemma for regexmid 4596. 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 4243	. . . 4 ⊢ {∅} ∈ V
2	1	prid2 3745	. . 3 ⊢ {∅} ∈ {∅, {∅}}
3		eqid 2206	. . . 4 ⊢ {∅} = {∅}
4	3	orci 733	. . 3 ⊢ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑))
5		eqeq1 2213	. . . . 5 ⊢ (𝑥 = {∅} → (𝑥 = {∅} ↔ {∅} = {∅}))
6		eqeq1 2213	. . . . . 6 ⊢ (𝑥 = {∅} → (𝑥 = ∅ ↔ {∅} = ∅))
7	6	anbi1d 465	. . . . 5 ⊢ (𝑥 = {∅} → ((𝑥 = ∅ ∧ 𝜑) ↔ ({∅} = ∅ ∧ 𝜑)))
8	5, 7	orbi12d 795	. . . 4 ⊢ (𝑥 = {∅} → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑))))
9		regexmidlemm.a	. . . 4 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
10	8, 9	elrab2 2936	. . 3 ⊢ ({∅} ∈ 𝐴 ↔ ({∅} ∈ {∅, {∅}} ∧ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑))))
11	2, 4, 10	mpbir2an 945	. 2 ⊢ {∅} ∈ 𝐴
12		elex2 2790	. 2 ⊢ ({∅} ∈ 𝐴 → ∃𝑦 𝑦 ∈ 𝐴)
13	11, 12	ax-mp 5	1 ⊢ ∃𝑦 𝑦 ∈ 𝐴

Syntax hints:   ∧ wa 104   ∨ wo 710   = wceq 1373  ∃wex 1516   ∈ wcel 2177  {crab 2489  ∅c0 3464  {csn 3638  {cpr 3639

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-nul 4181  ax-pow 4229

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rab 2494  df-v 2775  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645

This theorem is referenced by:  regexmid  4596  reg2exmid  4597  reg3exmid  4641  nnregexmid  4682