Theorem nnregexmid 4598

Description: If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4512 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6467 or nntri3or 6461), sets of natural numbers are a different animal. (Contributed by Jim Kingdon, 6-Sep-2019.)

Hypothesis

Ref	Expression
nnregexmid.1	⊢ ((𝑥 ⊆ ω ∧ ∃𝑦 𝑦 ∈ 𝑥) → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))

Assertion

Ref	Expression
nnregexmid	⊢ (𝜑 ∨ ¬ 𝜑)

Distinct variable group:   𝜑,𝑥,𝑦,𝑧

Proof of Theorem nnregexmid

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 3227	. . . 4 ⊢ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ⊆ {∅, {∅}}
2		peano1 4571	. . . . 5 ⊢ ∅ ∈ ω
3		suc0 4389	. . . . . 6 ⊢ suc ∅ = {∅}
4		peano2 4572	. . . . . . 7 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
5	2, 4	ax-mp 5	. . . . . 6 ⊢ suc ∅ ∈ ω
6	3, 5	eqeltrri 2240	. . . . 5 ⊢ {∅} ∈ ω
7		prssi 3731	. . . . 5 ⊢ ((∅ ∈ ω ∧ {∅} ∈ ω) → {∅, {∅}} ⊆ ω)
8	2, 6, 7	mp2an 423	. . . 4 ⊢ {∅, {∅}} ⊆ ω
9	1, 8	sstri 3151	. . 3 ⊢ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ⊆ ω
10		eqid 2165	. . . 4 ⊢ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}
11	10	regexmidlemm 4509	. . 3 ⊢ ∃𝑦 𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}
12		pp0ex 4168	. . . . 5 ⊢ {∅, {∅}} ∈ V
13	12	rabex 4126	. . . 4 ⊢ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ∈ V
14		sseq1 3165	. . . . . 6 ⊢ (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (𝑥 ⊆ ω ↔ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ⊆ ω))
15		eleq2 2230	. . . . . . 7 ⊢ (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}))
16	15	exbidv 1813	. . . . . 6 ⊢ (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (∃𝑦 𝑦 ∈ 𝑥 ↔ ∃𝑦 𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}))
17	14, 16	anbi12d 465	. . . . 5 ⊢ (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → ((𝑥 ⊆ ω ∧ ∃𝑦 𝑦 ∈ 𝑥) ↔ ({𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ⊆ ω ∧ ∃𝑦 𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))})))
18		eleq2 2230	. . . . . . . . . 10 ⊢ (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}))
19	18	notbid 657	. . . . . . . . 9 ⊢ (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (¬ 𝑧 ∈ 𝑥 ↔ ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}))
20	19	imbi2d 229	. . . . . . . 8 ⊢ (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → ((𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥) ↔ (𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))})))
21	20	albidv 1812	. . . . . . 7 ⊢ (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥) ↔ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))})))
22	15, 21	anbi12d 465	. . . . . 6 ⊢ (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → ((𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)) ↔ (𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}))))
23	22	exbidv 1813	. . . . 5 ⊢ (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)) ↔ ∃𝑦(𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}))))
24	17, 23	imbi12d 233	. . . 4 ⊢ (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (((𝑥 ⊆ ω ∧ ∃𝑦 𝑦 ∈ 𝑥) → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))) ↔ (({𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ⊆ ω ∧ ∃𝑦 𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}) → ∃𝑦(𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))})))))
25		nnregexmid.1	. . . 4 ⊢ ((𝑥 ⊆ ω ∧ ∃𝑦 𝑦 ∈ 𝑥) → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))
26	13, 24, 25	vtocl 2780	. . 3 ⊢ (({𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ⊆ ω ∧ ∃𝑦 𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}) → ∃𝑦(𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))})))
27	9, 11, 26	mp2an 423	. 2 ⊢ ∃𝑦(𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}))
28	10	regexmidlem1 4510	. 2 ⊢ (∃𝑦(𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))})) → (𝜑 ∨ ¬ 𝜑))
29	27, 28	ax-mp 5	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 698  ∀wal 1341   = wceq 1343  ∃wex 1480   ∈ wcel 2136  {crab 2448   ⊆ wss 3116  ∅c0 3409  {csn 3576  {cpr 3577  suc csuc 4343  ωcom 4567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-uni 3790  df-int 3825  df-suc 4349  df-iom 4568

This theorem is referenced by: (None)