Theorem regexmidlem1 4544

Description: Lemma for regexmid 4546. 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 2194	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 = {∅} ↔ 𝑦 = {∅}))
2		eqeq1 2194	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
3	2	anbi1d 465	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑥 = ∅ ∧ 𝜑) ↔ (𝑦 = ∅ ∧ 𝜑)))
4	1, 3	orbi12d 794	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ (𝑦 = {∅} ∨ (𝑦 = ∅ ∧ 𝜑))))
5		regexmidlemm.a	. . . . . 6 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
6	4, 5	elrab2 2908	. . . . 5 ⊢ (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ {∅, {∅}} ∧ (𝑦 = {∅} ∨ (𝑦 = ∅ ∧ 𝜑))))
7	6	simprbi 275	. . . 4 ⊢ (𝑦 ∈ 𝐴 → (𝑦 = {∅} ∨ (𝑦 = ∅ ∧ 𝜑)))
8		0ex 4142	. . . . . . . . 9 ⊢ ∅ ∈ V
9	8	snid 3635	. . . . . . . 8 ⊢ ∅ ∈ {∅}
10		eleq2 2251	. . . . . . . 8 ⊢ (𝑦 = {∅} → (∅ ∈ 𝑦 ↔ ∅ ∈ {∅}))
11	9, 10	mpbiri 168	. . . . . . 7 ⊢ (𝑦 = {∅} → ∅ ∈ 𝑦)
12		eleq1 2250	. . . . . . . . 9 ⊢ (𝑧 = ∅ → (𝑧 ∈ 𝑦 ↔ ∅ ∈ 𝑦))
13		eleq1 2250	. . . . . . . . . 10 ⊢ (𝑧 = ∅ → (𝑧 ∈ 𝐴 ↔ ∅ ∈ 𝐴))
14	13	notbid 668	. . . . . . . . 9 ⊢ (𝑧 = ∅ → (¬ 𝑧 ∈ 𝐴 ↔ ¬ ∅ ∈ 𝐴))
15	12, 14	imbi12d 234	. . . . . . . 8 ⊢ (𝑧 = ∅ → ((𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴) ↔ (∅ ∈ 𝑦 → ¬ ∅ ∈ 𝐴)))
16	8, 15	spcv 2843	. . . . . . 7 ⊢ (∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴) → (∅ ∈ 𝑦 → ¬ ∅ ∈ 𝐴))
17	11, 16	syl5com 29	. . . . . 6 ⊢ (𝑦 = {∅} → (∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴) → ¬ ∅ ∈ 𝐴))
18	8	prid1 3710	. . . . . . . . . 10 ⊢ ∅ ∈ {∅, {∅}}
19		eqeq1 2194	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑥 = {∅} ↔ ∅ = {∅}))
20		eqeq1 2194	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
21	20	anbi1d 465	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → ((𝑥 = ∅ ∧ 𝜑) ↔ (∅ = ∅ ∧ 𝜑)))
22	19, 21	orbi12d 794	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑))))
23	22, 5	elrab2 2908	. . . . . . . . . 10 ⊢ (∅ ∈ 𝐴 ↔ (∅ ∈ {∅, {∅}} ∧ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑))))
24	18, 23	mpbiran 941	. . . . . . . . 9 ⊢ (∅ ∈ 𝐴 ↔ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑)))
25		pm2.46 740	. . . . . . . . 9 ⊢ (¬ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑)) → ¬ (∅ = ∅ ∧ 𝜑))
26	24, 25	sylnbi 679	. . . . . . . 8 ⊢ (¬ ∅ ∈ 𝐴 → ¬ (∅ = ∅ ∧ 𝜑))
27		eqid 2187	. . . . . . . . 9 ⊢ ∅ = ∅
28	27	biantrur 303	. . . . . . . 8 ⊢ (𝜑 ↔ (∅ = ∅ ∧ 𝜑))
29	26, 28	sylnibr 678	. . . . . . 7 ⊢ (¬ ∅ ∈ 𝐴 → ¬ 𝜑)
30	29	olcd 735	. . . . . 6 ⊢ (¬ ∅ ∈ 𝐴 → (𝜑 ∨ ¬ 𝜑))
31	17, 30	syl6 33	. . . . 5 ⊢ (𝑦 = {∅} → (∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴) → (𝜑 ∨ ¬ 𝜑)))
32		orc 713	. . . . . . 7 ⊢ (𝜑 → (𝜑 ∨ ¬ 𝜑))
33	32	adantl 277	. . . . . 6 ⊢ ((𝑦 = ∅ ∧ 𝜑) → (𝜑 ∨ ¬ 𝜑))
34	33	a1d 22	. . . . 5 ⊢ ((𝑦 = ∅ ∧ 𝜑) → (∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴) → (𝜑 ∨ ¬ 𝜑)))
35	31, 34	jaoi 717	. . . 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 124	. 2 ⊢ ((𝑦 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴)) → (𝜑 ∨ ¬ 𝜑))
38	37	exlimiv 1608	1 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴)) → (𝜑 ∨ ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 709  ∀wal 1361   = wceq 1363  ∃wex 1502   ∈ wcel 2158  {crab 2469  ∅c0 3434  {csn 3604  {cpr 3605

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169  ax-nul 4141

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-nul 3435  df-sn 3610  df-pr 3611

This theorem is referenced by:  regexmid  4546  nnregexmid  4632