Theorem regexmidlemm 4674

Description: Lemma for regexmid 4677. 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 4320	. . . 4 ⊢ {∅} ∈ V
2	1	prid2 3814	. . 3 ⊢ {∅} ∈ {∅, {∅}}
3		eqid 2238	. . . 4 ⊢ {∅} = {∅}
4	3	orci 743	. . 3 ⊢ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑))
5		eqeq1 2245	. . . . 5 ⊢ (𝑥 = {∅} → (𝑥 = {∅} ↔ {∅} = {∅}))
6		eqeq1 2245	. . . . . 6 ⊢ (𝑥 = {∅} → (𝑥 = ∅ ↔ {∅} = ∅))
7	6	anbi1d 469	. . . . 5 ⊢ (𝑥 = {∅} → ((𝑥 = ∅ ∧ 𝜑) ↔ ({∅} = ∅ ∧ 𝜑)))
8	5, 7	orbi12d 805	. . . 4 ⊢ (𝑥 = {∅} → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑))))
9		regexmidlemm.a	. . . 4 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
10	8, 9	elrab2 2985	. . 3 ⊢ ({∅} ∈ 𝐴 ↔ ({∅} ∈ {∅, {∅}} ∧ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑))))
11	2, 4, 10	mpbir2an 955	. 2 ⊢ {∅} ∈ 𝐴
12		elex2 2838	. 2 ⊢ ({∅} ∈ 𝐴 → ∃𝑦 𝑦 ∈ 𝐴)
13	11, 12	ax-mp 5	1 ⊢ ∃𝑦 𝑦 ∈ 𝐴

Syntax hints:   ∧ wa 104   ∨ wo 720   = wceq 1402  ∃wex 1545   ∈ wcel 2209  {crab 2532  ∅c0 3520  {csn 3705  {cpr 3706

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712

This theorem is referenced by:  regexmid  4677  reg2exmid  4678  reg3exmid  4722  nnregexmid  4763