Theorem reg3exmid 4701

Description: If any inhabited set satisfying df-wetr 4454 for E has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Oct-2021.)

Hypothesis

Ref	Expression
reg3exmid.1	⊢ (( E We 𝑧 ∧ ∃𝑤 𝑤 ∈ 𝑧) → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 𝑥 ⊆ 𝑦)

Assertion

Ref	Expression
reg3exmid	⊢ (𝜑 ∨ ¬ 𝜑)

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

Proof of Theorem reg3exmid

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2232	. . 3 ⊢ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} = {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}
2	1	regexmidlemm 4653	. 2 ⊢ ∃𝑤 𝑤 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}
3	1	reg3exmidlemwe 4700	. . 3 ⊢ E We {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}
4		pp0ex 4301	. . . . 5 ⊢ {∅, {∅}} ∈ V
5	4	rabex 4255	. . . 4 ⊢ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} ∈ V
6		weeq2 4477	. . . . . 6 ⊢ (𝑧 = {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} → ( E We 𝑧 ↔ E We {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}))
7		eleq2 2296	. . . . . . 7 ⊢ (𝑧 = {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}))
8	7	exbidv 1874	. . . . . 6 ⊢ (𝑧 = {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} → (∃𝑤 𝑤 ∈ 𝑧 ↔ ∃𝑤 𝑤 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}))
9	6, 8	anbi12d 473	. . . . 5 ⊢ (𝑧 = {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} → (( E We 𝑧 ∧ ∃𝑤 𝑤 ∈ 𝑧) ↔ ( E We {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} ∧ ∃𝑤 𝑤 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))})))
10		raleq 2740	. . . . . 6 ⊢ (𝑧 = {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} → (∀𝑦 ∈ 𝑧 𝑥 ⊆ 𝑦 ↔ ∀𝑦 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}𝑥 ⊆ 𝑦))
11	10	rexeqbi1dv 2753	. . . . 5 ⊢ (𝑧 = {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} → (∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 𝑥 ⊆ 𝑦 ↔ ∃𝑥 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}∀𝑦 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}𝑥 ⊆ 𝑦))
12	9, 11	imbi12d 234	. . . 4 ⊢ (𝑧 = {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} → ((( E We 𝑧 ∧ ∃𝑤 𝑤 ∈ 𝑧) → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 𝑥 ⊆ 𝑦) ↔ (( E We {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} ∧ ∃𝑤 𝑤 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}) → ∃𝑥 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}∀𝑦 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}𝑥 ⊆ 𝑦)))
13		reg3exmid.1	. . . 4 ⊢ (( E We 𝑧 ∧ ∃𝑤 𝑤 ∈ 𝑧) → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 𝑥 ⊆ 𝑦)
14	5, 12, 13	vtocl 2868	. . 3 ⊢ (( E We {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} ∧ ∃𝑤 𝑤 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}) → ∃𝑥 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}∀𝑦 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}𝑥 ⊆ 𝑦)
15	3, 14	mpan 424	. 2 ⊢ (∃𝑤 𝑤 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} → ∃𝑥 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}∀𝑦 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}𝑥 ⊆ 𝑦)
16	1	reg2exmidlema 4655	. 2 ⊢ (∃𝑥 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}∀𝑦 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}𝑥 ⊆ 𝑦 → (𝜑 ∨ ¬ 𝜑))
17	2, 15, 16	mp2b 8	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 716   = wceq 1398  ∃wex 1541   ∈ wcel 2203  ∀wral 2520  ∃wrex 2521  {crab 2524   ⊆ wss 3210  ∅c0 3507  {csn 3688  {cpr 3689   E cep 4407   We wwe 4450

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 2206  ax-ext 2214  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-setind 4658

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  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-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-br 4109  df-opab 4171  df-eprel 4409  df-frfor 4451  df-frind 4452  df-wetr 4454

This theorem is referenced by: (None)