Theorem regexmidlem1 4386

Description: Lemma for regexmid 4388. If 𝐴 has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2019.)

Hypothesis

Ref	Expression
regexmidlemm.a	⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}

Assertion

Ref	Expression
regexmidlem1	⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴)) → (𝜑 ∨ ¬ 𝜑))

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

Allowed substitution hints:   𝜑(𝑧)   𝐴(𝑥)

Proof of Theorem regexmidlem1

Step	Hyp	Ref	Expression
1		eqeq1 2106	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 = {∅} ↔ 𝑦 = {∅}))
2		eqeq1 2106	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
3	2	anbi1d 456	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑥 = ∅ ∧ 𝜑) ↔ (𝑦 = ∅ ∧ 𝜑)))
4	1, 3	orbi12d 748	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ (𝑦 = {∅} ∨ (𝑦 = ∅ ∧ 𝜑))))
5		regexmidlemm.a	. . . . . 6 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
6	4, 5	elrab2 2796	. . . . 5 ⊢ (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ {∅, {∅}} ∧ (𝑦 = {∅} ∨ (𝑦 = ∅ ∧ 𝜑))))
7	6	simprbi 271	. . . 4 ⊢ (𝑦 ∈ 𝐴 → (𝑦 = {∅} ∨ (𝑦 = ∅ ∧ 𝜑)))
8		0ex 3995	. . . . . . . . 9 ⊢ ∅ ∈ V
9	8	snid 3503	. . . . . . . 8 ⊢ ∅ ∈ {∅}
10		eleq2 2163	. . . . . . . 8 ⊢ (𝑦 = {∅} → (∅ ∈ 𝑦 ↔ ∅ ∈ {∅}))
11	9, 10	mpbiri 167	. . . . . . 7 ⊢ (𝑦 = {∅} → ∅ ∈ 𝑦)
12		eleq1 2162	. . . . . . . . 9 ⊢ (𝑧 = ∅ → (𝑧 ∈ 𝑦 ↔ ∅ ∈ 𝑦))
13		eleq1 2162	. . . . . . . . . 10 ⊢ (𝑧 = ∅ → (𝑧 ∈ 𝐴 ↔ ∅ ∈ 𝐴))
14	13	notbid 633	. . . . . . . . 9 ⊢ (𝑧 = ∅ → (¬ 𝑧 ∈ 𝐴 ↔ ¬ ∅ ∈ 𝐴))
15	12, 14	imbi12d 233	. . . . . . . 8 ⊢ (𝑧 = ∅ → ((𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴) ↔ (∅ ∈ 𝑦 → ¬ ∅ ∈ 𝐴)))
16	8, 15	spcv 2734	. . . . . . 7 ⊢ (∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴) → (∅ ∈ 𝑦 → ¬ ∅ ∈ 𝐴))
17	11, 16	syl5com 29	. . . . . 6 ⊢ (𝑦 = {∅} → (∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴) → ¬ ∅ ∈ 𝐴))
18	8	prid1 3576	. . . . . . . . . 10 ⊢ ∅ ∈ {∅, {∅}}
19		eqeq1 2106	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑥 = {∅} ↔ ∅ = {∅}))
20		eqeq1 2106	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
21	20	anbi1d 456	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → ((𝑥 = ∅ ∧ 𝜑) ↔ (∅ = ∅ ∧ 𝜑)))
22	19, 21	orbi12d 748	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑))))
23	22, 5	elrab2 2796	. . . . . . . . . 10 ⊢ (∅ ∈ 𝐴 ↔ (∅ ∈ {∅, {∅}} ∧ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑))))
24	18, 23	mpbiran 892	. . . . . . . . 9 ⊢ (∅ ∈ 𝐴 ↔ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑)))
25		pm2.46 699	. . . . . . . . 9 ⊢ (¬ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑)) → ¬ (∅ = ∅ ∧ 𝜑))
26	24, 25	sylnbi 644	. . . . . . . 8 ⊢ (¬ ∅ ∈ 𝐴 → ¬ (∅ = ∅ ∧ 𝜑))
27		eqid 2100	. . . . . . . . 9 ⊢ ∅ = ∅
28	27	biantrur 299	. . . . . . . 8 ⊢ (𝜑 ↔ (∅ = ∅ ∧ 𝜑))
29	26, 28	sylnibr 643	. . . . . . 7 ⊢ (¬ ∅ ∈ 𝐴 → ¬ 𝜑)
30	29	olcd 694	. . . . . 6 ⊢ (¬ ∅ ∈ 𝐴 → (𝜑 ∨ ¬ 𝜑))
31	17, 30	syl6 33	. . . . 5 ⊢ (𝑦 = {∅} → (∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴) → (𝜑 ∨ ¬ 𝜑)))
32		orc 674	. . . . . . 7 ⊢ (𝜑 → (𝜑 ∨ ¬ 𝜑))
33	32	adantl 273	. . . . . 6 ⊢ ((𝑦 = ∅ ∧ 𝜑) → (𝜑 ∨ ¬ 𝜑))
34	33	a1d 22	. . . . 5 ⊢ ((𝑦 = ∅ ∧ 𝜑) → (∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴) → (𝜑 ∨ ¬ 𝜑)))
35	31, 34	jaoi 677	. . . 4 ⊢ ((𝑦 = {∅} ∨ (𝑦 = ∅ ∧ 𝜑)) → (∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴) → (𝜑 ∨ ¬ 𝜑)))
36	7, 35	syl 14	. . 3 ⊢ (𝑦 ∈ 𝐴 → (∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴) → (𝜑 ∨ ¬ 𝜑)))
37	36	imp 123	. 2 ⊢ ((𝑦 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴)) → (𝜑 ∨ ¬ 𝜑))
38	37	exlimiv 1545	1 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴)) → (𝜑 ∨ ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 670  ∀wal 1297   = 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-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-nul 3994

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-nul 3311  df-sn 3480  df-pr 3481

This theorem is referenced by:  regexmid  4388  nnregexmid  4472