Theorem reg3exmid 4616

Description: If any inhabited set satisfying df-wetr 4369 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 2196	. . 3 ⊢ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} = {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}
2	1	regexmidlemm 4568	. 2 ⊢ ∃𝑤 𝑤 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}
3	1	reg3exmidlemwe 4615	. . 3 ⊢ E We {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}
4		pp0ex 4222	. . . . 5 ⊢ {∅, {∅}} ∈ V
5	4	rabex 4177	. . . 4 ⊢ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} ∈ V
6		weeq2 4392	. . . . . 6 ⊢ (𝑧 = {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} → ( E We 𝑧 ↔ E We {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}))
7		eleq2 2260	. . . . . . 7 ⊢ (𝑧 = {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}))
8	7	exbidv 1839	. . . . . 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 2693	. . . . . 6 ⊢ (𝑧 = {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} → (∀𝑦 ∈ 𝑧 𝑥 ⊆ 𝑦 ↔ ∀𝑦 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}𝑥 ⊆ 𝑦))
11	10	rexeqbi1dv 2706	. . . . 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 2818	. . 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 4570	. 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 709   = wceq 1364  ∃wex 1506   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  {crab 2479   ⊆ wss 3157  ∅c0 3450  {csn 3622  {cpr 3623   E cep 4322   We wwe 4365

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-eprel 4324  df-frfor 4366  df-frind 4367  df-wetr 4369

This theorem is referenced by: (None)