Theorem regexmidlemm 4656

Description: Lemma for regexmid 4659. 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 4303	. . . 4 ⊢ {∅} ∈ V
2	1	prid2 3800	. . 3 ⊢ {∅} ∈ {∅, {∅}}
3		eqid 2234	. . . 4 ⊢ {∅} = {∅}
4	3	orci 739	. . 3 ⊢ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑))
5		eqeq1 2241	. . . . 5 ⊢ (𝑥 = {∅} → (𝑥 = {∅} ↔ {∅} = {∅}))
6		eqeq1 2241	. . . . . 6 ⊢ (𝑥 = {∅} → (𝑥 = ∅ ↔ {∅} = ∅))
7	6	anbi1d 465	. . . . 5 ⊢ (𝑥 = {∅} → ((𝑥 = ∅ ∧ 𝜑) ↔ ({∅} = ∅ ∧ 𝜑)))
8	5, 7	orbi12d 801	. . . 4 ⊢ (𝑥 = {∅} → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑))))
9		regexmidlemm.a	. . . 4 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
10	8, 9	elrab2 2978	. . 3 ⊢ ({∅} ∈ 𝐴 ↔ ({∅} ∈ {∅, {∅}} ∧ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑))))
11	2, 4, 10	mpbir2an 951	. 2 ⊢ {∅} ∈ 𝐴
12		elex2 2832	. 2 ⊢ ({∅} ∈ 𝐴 → ∃𝑦 𝑦 ∈ 𝐴)
13	11, 12	ax-mp 5	1 ⊢ ∃𝑦 𝑦 ∈ 𝐴

Syntax hints:   ∧ wa 104   ∨ wo 716   = wceq 1398  ∃wex 1541   ∈ wcel 2205  {crab 2526  ∅c0 3510  {csn 3691  {cpr 3692

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rab 2531  df-v 2817  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698

This theorem is referenced by:  regexmid  4659  reg2exmid  4660  reg3exmid  4704  nnregexmid  4745