Theorem reg2exmidlema 4655

Description: Lemma for reg2exmid 4657. 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 529	. . . . . . 7 ⊢ (((𝑢 ∈ 𝐴 ∧ ∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣) ∧ 𝑢 = {∅}) → ∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣)
2		sseq1 3260	. . . . . . . . 9 ⊢ (𝑢 = {∅} → (𝑢 ⊆ 𝑣 ↔ {∅} ⊆ 𝑣))
3	2	ralbidv 2542	. . . . . . . 8 ⊢ (𝑢 = {∅} → (∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣 ↔ ∀𝑣 ∈ 𝐴 {∅} ⊆ 𝑣))
4	3	adantl 277	. . . . . . 7 ⊢ (((𝑢 ∈ 𝐴 ∧ ∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣) ∧ 𝑢 = {∅}) → (∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣 ↔ ∀𝑣 ∈ 𝐴 {∅} ⊆ 𝑣))
5	1, 4	mpbid 147	. . . . . 6 ⊢ (((𝑢 ∈ 𝐴 ∧ ∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣) ∧ 𝑢 = {∅}) → ∀𝑣 ∈ 𝐴 {∅} ⊆ 𝑣)
6		0ex 4236	. . . . . . . 8 ⊢ ∅ ∈ V
7	6	snss 3828	. . . . . . 7 ⊢ (∅ ∈ 𝑣 ↔ {∅} ⊆ 𝑣)
8	7	ralbii 2548	. . . . . 6 ⊢ (∀𝑣 ∈ 𝐴 ∅ ∈ 𝑣 ↔ ∀𝑣 ∈ 𝐴 {∅} ⊆ 𝑣)
9	5, 8	sylibr 134	. . . . 5 ⊢ (((𝑢 ∈ 𝐴 ∧ ∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣) ∧ 𝑢 = {∅}) → ∀𝑣 ∈ 𝐴 ∅ ∈ 𝑣)
10		noel 3511	. . . . . 6 ⊢ ¬ ∅ ∈ ∅
11		eqid 2232	. . . . . . . . . . . 12 ⊢ ∅ = ∅
12	11	jctl 314	. . . . . . . . . . 11 ⊢ (𝜑 → (∅ = ∅ ∧ 𝜑))
13	12	olcd 742	. . . . . . . . . 10 ⊢ (𝜑 → (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑)))
14	6	prid1 3796	. . . . . . . . . 10 ⊢ ∅ ∈ {∅, {∅}}
15	13, 14	jctil 312	. . . . . . . . 9 ⊢ (𝜑 → (∅ ∈ {∅, {∅}} ∧ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑))))
16		eqeq1 2239	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑥 = {∅} ↔ ∅ = {∅}))
17		eqeq1 2239	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
18	17	anbi1d 465	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → ((𝑥 = ∅ ∧ 𝜑) ↔ (∅ = ∅ ∧ 𝜑)))
19	16, 18	orbi12d 801	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑))))
20		regexmidlemm.a	. . . . . . . . . 10 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
21	19, 20	elrab2 2975	. . . . . . . . 9 ⊢ (∅ ∈ 𝐴 ↔ (∅ ∈ {∅, {∅}} ∧ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑))))
22	15, 21	sylibr 134	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ 𝐴)
23		eleq2 2296	. . . . . . . . 9 ⊢ (𝑣 = ∅ → (∅ ∈ 𝑣 ↔ ∅ ∈ ∅))
24	23	rspcv 2916	. . . . . . . 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 670	. . . . 5 ⊢ (∀𝑣 ∈ 𝐴 ∅ ∈ 𝑣 → ¬ 𝜑)
28	9, 27	syl 14	. . . 4 ⊢ (((𝑢 ∈ 𝐴 ∧ ∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣) ∧ 𝑢 = {∅}) → ¬ 𝜑)
29	28	olcd 742	. . 3 ⊢ (((𝑢 ∈ 𝐴 ∧ ∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣) ∧ 𝑢 = {∅}) → (𝜑 ∨ ¬ 𝜑))
30		simprr 533	. . . 4 ⊢ (((𝑢 ∈ 𝐴 ∧ ∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣) ∧ (𝑢 = ∅ ∧ 𝜑)) → 𝜑)
31	30	orcd 741	. . 3 ⊢ (((𝑢 ∈ 𝐴 ∧ ∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣) ∧ (𝑢 = ∅ ∧ 𝜑)) → (𝜑 ∨ ¬ 𝜑))
32		eqeq1 2239	. . . . . . 7 ⊢ (𝑥 = 𝑢 → (𝑥 = {∅} ↔ 𝑢 = {∅}))
33		eqeq1 2239	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → (𝑥 = ∅ ↔ 𝑢 = ∅))
34	33	anbi1d 465	. . . . . . 7 ⊢ (𝑥 = 𝑢 → ((𝑥 = ∅ ∧ 𝜑) ↔ (𝑢 = ∅ ∧ 𝜑)))
35	32, 34	orbi12d 801	. . . . . 6 ⊢ (𝑥 = 𝑢 → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))))
36	35, 20	elrab2 2975	. . . . 5 ⊢ (𝑢 ∈ 𝐴 ↔ (𝑢 ∈ {∅, {∅}} ∧ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))))
37	36	simprbi 275	. . . 4 ⊢ (𝑢 ∈ 𝐴 → (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑)))
38	37	adantr 276	. . 3 ⊢ ((𝑢 ∈ 𝐴 ∧ ∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣) → (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑)))
39	29, 31, 38	mpjaodan 806	. 2 ⊢ ((𝑢 ∈ 𝐴 ∧ ∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣) → (𝜑 ∨ ¬ 𝜑))
40	39	rexlimiva 2655	1 ⊢ (∃𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣 → (𝜑 ∨ ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   = wceq 1398   ∈ wcel 2203  ∀wral 2520  ∃wrex 2521  {crab 2524   ⊆ wss 3210  ∅c0 3507  {csn 3688  {cpr 3689

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-ext 2214  ax-nul 4235

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-sn 3694  df-pr 3695

This theorem is referenced by:  reg2exmid  4657  reg3exmid  4701