Theorem reg2exmidlema 4407

Description: Lemma for reg2exmid 4409. If 𝐴 has a minimal element (expressed by ⊆), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.)

Hypothesis

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

Assertion

Ref	Expression
reg2exmidlema	⊢ (∃𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣 → (𝜑 ∨ ¬ 𝜑))

Distinct variable groups:   𝜑,𝑥   𝑣,𝐴   𝜑,𝑢,𝑥   𝑣,𝑢

Allowed substitution hints:   𝜑(𝑣)   𝐴(𝑥,𝑢)

Proof of Theorem reg2exmidlema

Step	Hyp	Ref	Expression
1		simplr 502	. . . . . . 7 ⊢ (((𝑢 ∈ 𝐴 ∧ ∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣) ∧ 𝑢 = {∅}) → ∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣)
2		sseq1 3084	. . . . . . . . 9 ⊢ (𝑢 = {∅} → (𝑢 ⊆ 𝑣 ↔ {∅} ⊆ 𝑣))
3	2	ralbidv 2409	. . . . . . . 8 ⊢ (𝑢 = {∅} → (∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣 ↔ ∀𝑣 ∈ 𝐴 {∅} ⊆ 𝑣))
4	3	adantl 273	. . . . . . 7 ⊢ (((𝑢 ∈ 𝐴 ∧ ∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣) ∧ 𝑢 = {∅}) → (∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣 ↔ ∀𝑣 ∈ 𝐴 {∅} ⊆ 𝑣))
5	1, 4	mpbid 146	. . . . . 6 ⊢ (((𝑢 ∈ 𝐴 ∧ ∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣) ∧ 𝑢 = {∅}) → ∀𝑣 ∈ 𝐴 {∅} ⊆ 𝑣)
6		0ex 4013	. . . . . . . 8 ⊢ ∅ ∈ V
7	6	snss 3613	. . . . . . 7 ⊢ (∅ ∈ 𝑣 ↔ {∅} ⊆ 𝑣)
8	7	ralbii 2413	. . . . . 6 ⊢ (∀𝑣 ∈ 𝐴 ∅ ∈ 𝑣 ↔ ∀𝑣 ∈ 𝐴 {∅} ⊆ 𝑣)
9	5, 8	sylibr 133	. . . . 5 ⊢ (((𝑢 ∈ 𝐴 ∧ ∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣) ∧ 𝑢 = {∅}) → ∀𝑣 ∈ 𝐴 ∅ ∈ 𝑣)
10		noel 3331	. . . . . 6 ⊢ ¬ ∅ ∈ ∅
11		eqid 2113	. . . . . . . . . . . 12 ⊢ ∅ = ∅
12	11	jctl 310	. . . . . . . . . . 11 ⊢ (𝜑 → (∅ = ∅ ∧ 𝜑))
13	12	olcd 706	. . . . . . . . . 10 ⊢ (𝜑 → (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑)))
14	6	prid1 3593	. . . . . . . . . 10 ⊢ ∅ ∈ {∅, {∅}}
15	13, 14	jctil 308	. . . . . . . . 9 ⊢ (𝜑 → (∅ ∈ {∅, {∅}} ∧ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑))))
16		eqeq1 2119	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑥 = {∅} ↔ ∅ = {∅}))
17		eqeq1 2119	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
18	17	anbi1d 458	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → ((𝑥 = ∅ ∧ 𝜑) ↔ (∅ = ∅ ∧ 𝜑)))
19	16, 18	orbi12d 765	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑))))
20		regexmidlemm.a	. . . . . . . . . 10 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
21	19, 20	elrab2 2810	. . . . . . . . 9 ⊢ (∅ ∈ 𝐴 ↔ (∅ ∈ {∅, {∅}} ∧ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑))))
22	15, 21	sylibr 133	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ 𝐴)
23		eleq2 2176	. . . . . . . . 9 ⊢ (𝑣 = ∅ → (∅ ∈ 𝑣 ↔ ∅ ∈ ∅))
24	23	rspcv 2754	. . . . . . . 8 ⊢ (∅ ∈ 𝐴 → (∀𝑣 ∈ 𝐴 ∅ ∈ 𝑣 → ∅ ∈ ∅))
25	22, 24	syl 14	. . . . . . 7 ⊢ (𝜑 → (∀𝑣 ∈ 𝐴 ∅ ∈ 𝑣 → ∅ ∈ ∅))
26	25	com12 30	. . . . . 6 ⊢ (∀𝑣 ∈ 𝐴 ∅ ∈ 𝑣 → (𝜑 → ∅ ∈ ∅))
27	10, 26	mtoi 636	. . . . 5 ⊢ (∀𝑣 ∈ 𝐴 ∅ ∈ 𝑣 → ¬ 𝜑)
28	9, 27	syl 14	. . . 4 ⊢ (((𝑢 ∈ 𝐴 ∧ ∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣) ∧ 𝑢 = {∅}) → ¬ 𝜑)
29	28	olcd 706	. . 3 ⊢ (((𝑢 ∈ 𝐴 ∧ ∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣) ∧ 𝑢 = {∅}) → (𝜑 ∨ ¬ 𝜑))
30		simprr 504	. . . 4 ⊢ (((𝑢 ∈ 𝐴 ∧ ∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣) ∧ (𝑢 = ∅ ∧ 𝜑)) → 𝜑)
31	30	orcd 705	. . 3 ⊢ (((𝑢 ∈ 𝐴 ∧ ∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣) ∧ (𝑢 = ∅ ∧ 𝜑)) → (𝜑 ∨ ¬ 𝜑))
32		eqeq1 2119	. . . . . . 7 ⊢ (𝑥 = 𝑢 → (𝑥 = {∅} ↔ 𝑢 = {∅}))
33		eqeq1 2119	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → (𝑥 = ∅ ↔ 𝑢 = ∅))
34	33	anbi1d 458	. . . . . . 7 ⊢ (𝑥 = 𝑢 → ((𝑥 = ∅ ∧ 𝜑) ↔ (𝑢 = ∅ ∧ 𝜑)))
35	32, 34	orbi12d 765	. . . . . 6 ⊢ (𝑥 = 𝑢 → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))))
36	35, 20	elrab2 2810	. . . . 5 ⊢ (𝑢 ∈ 𝐴 ↔ (𝑢 ∈ {∅, {∅}} ∧ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))))
37	36	simprbi 271	. . . 4 ⊢ (𝑢 ∈ 𝐴 → (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑)))
38	37	adantr 272	. . 3 ⊢ ((𝑢 ∈ 𝐴 ∧ ∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣) → (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑)))
39	29, 31, 38	mpjaodan 770	. 2 ⊢ ((𝑢 ∈ 𝐴 ∧ ∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣) → (𝜑 ∨ ¬ 𝜑))
40	39	rexlimiva 2516	1 ⊢ (∃𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣 → (𝜑 ∨ ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 680   = wceq 1312   ∈ wcel 1461  ∀wral 2388  ∃wrex 2389  {crab 2392   ⊆ wss 3035  ∅c0 3327  {csn 3491  {cpr 3492

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 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-nul 4012

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-rab 2397  df-v 2657  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-sn 3497  df-pr 3498

This theorem is referenced by:  reg2exmid  4409  reg3exmid  4452