Theorem nnregexmid 4622

Description: If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4536 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6502 or nntri3or 6496), 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 3242	. . . 4 ⊢ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ⊆ {∅, {∅}}
2		peano1 4595	. . . . 5 ⊢ ∅ ∈ ω
3		suc0 4413	. . . . . 6 ⊢ suc ∅ = {∅}
4		peano2 4596	. . . . . . 7 ⊢ (∅ ∈ ω → suc ∅ ∈ ω)
5	2, 4	ax-mp 5	. . . . . 6 ⊢ suc ∅ ∈ ω
6	3, 5	eqeltrri 2251	. . . . 5 ⊢ {∅} ∈ ω
7		prssi 3752	. . . . 5 ⊢ ((∅ ∈ ω ∧ {∅} ∈ ω) → {∅, {∅}} ⊆ ω)
8	2, 6, 7	mp2an 426	. . . 4 ⊢ {∅, {∅}} ⊆ ω
9	1, 8	sstri 3166	. . 3 ⊢ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ⊆ ω
10		eqid 2177	. . . 4 ⊢ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}
11	10	regexmidlemm 4533	. . 3 ⊢ ∃𝑦 𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}
12		pp0ex 4191	. . . . 5 ⊢ {∅, {∅}} ∈ V
13	12	rabex 4149	. . . 4 ⊢ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ∈ V
14		sseq1 3180	. . . . . 6 ⊢ (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (𝑥 ⊆ ω ↔ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ⊆ ω))
15		eleq2 2241	. . . . . . 7 ⊢ (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}))
16	15	exbidv 1825	. . . . . 6 ⊢ (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (∃𝑦 𝑦 ∈ 𝑥 ↔ ∃𝑦 𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}))
17	14, 16	anbi12d 473	. . . . 5 ⊢ (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → ((𝑥 ⊆ ω ∧ ∃𝑦 𝑦 ∈ 𝑥) ↔ ({𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ⊆ ω ∧ ∃𝑦 𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))})))
18		eleq2 2241	. . . . . . . . . 10 ⊢ (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}))
19	18	notbid 667	. . . . . . . . 9 ⊢ (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (¬ 𝑧 ∈ 𝑥 ↔ ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}))
20	19	imbi2d 230	. . . . . . . 8 ⊢ (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → ((𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥) ↔ (𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))})))
21	20	albidv 1824	. . . . . . 7 ⊢ (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥) ↔ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))})))
22	15, 21	anbi12d 473	. . . . . 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 1825	. . . . 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 234	. . . 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 2793	. . 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 426	. 2 ⊢ ∃𝑦(𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}))
28	10	regexmidlem1 4534	. 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 104   ∨ wo 708  ∀wal 1351   = wceq 1353  ∃wex 1492   ∈ wcel 2148  {crab 2459   ⊆ wss 3131  ∅c0 3424  {csn 3594  {cpr 3595  suc csuc 4367  ωcom 4591

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-uni 3812  df-int 3847  df-suc 4373  df-iom 4592

This theorem is referenced by: (None)