Theorem regexmidlemm 4509

Description: Lemma for regexmid 4512. 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 4167	. . . 4 ⊢ {∅} ∈ V
2	1	prid2 3683	. . 3 ⊢ {∅} ∈ {∅, {∅}}
3		eqid 2165	. . . 4 ⊢ {∅} = {∅}
4	3	orci 721	. . 3 ⊢ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑))
5		eqeq1 2172	. . . . 5 ⊢ (𝑥 = {∅} → (𝑥 = {∅} ↔ {∅} = {∅}))
6		eqeq1 2172	. . . . . 6 ⊢ (𝑥 = {∅} → (𝑥 = ∅ ↔ {∅} = ∅))
7	6	anbi1d 461	. . . . 5 ⊢ (𝑥 = {∅} → ((𝑥 = ∅ ∧ 𝜑) ↔ ({∅} = ∅ ∧ 𝜑)))
8	5, 7	orbi12d 783	. . . 4 ⊢ (𝑥 = {∅} → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑))))
9		regexmidlemm.a	. . . 4 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
10	8, 9	elrab2 2885	. . 3 ⊢ ({∅} ∈ 𝐴 ↔ ({∅} ∈ {∅, {∅}} ∧ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑))))
11	2, 4, 10	mpbir2an 932	. 2 ⊢ {∅} ∈ 𝐴
12		elex2 2742	. 2 ⊢ ({∅} ∈ 𝐴 → ∃𝑦 𝑦 ∈ 𝐴)
13	11, 12	ax-mp 5	1 ⊢ ∃𝑦 𝑦 ∈ 𝐴

Syntax hints:   ∧ wa 103   ∨ wo 698   = wceq 1343  ∃wex 1480   ∈ wcel 2136  {crab 2448  ∅c0 3409  {csn 3576  {cpr 3577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583

This theorem is referenced by:  regexmid  4512  reg2exmid  4513  reg3exmid  4557  nnregexmid  4598