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Theorem nnregexmid 4580
Description: If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4494 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6446 or nntri3or 6440), sets of natural numbers are a different animal. (Contributed by Jim Kingdon, 6-Sep-2019.)
Hypothesis
Ref Expression
nnregexmid.1 ((𝑥 ⊆ ω ∧ ∃𝑦 𝑦𝑥) → ∃𝑦(𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥)))
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.
StepHypRef Expression
1 ssrab2 3213 . . . 4 {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ⊆ {∅, {∅}}
2 peano1 4553 . . . . 5 ∅ ∈ ω
3 suc0 4371 . . . . . 6 suc ∅ = {∅}
4 peano2 4554 . . . . . . 7 (∅ ∈ ω → suc ∅ ∈ ω)
52, 4ax-mp 5 . . . . . 6 suc ∅ ∈ ω
63, 5eqeltrri 2231 . . . . 5 {∅} ∈ ω
7 prssi 3714 . . . . 5 ((∅ ∈ ω ∧ {∅} ∈ ω) → {∅, {∅}} ⊆ ω)
82, 6, 7mp2an 423 . . . 4 {∅, {∅}} ⊆ ω
91, 8sstri 3137 . . 3 {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ⊆ ω
10 eqid 2157 . . . 4 {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}
1110regexmidlemm 4491 . . 3 𝑦 𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}
12 pp0ex 4150 . . . . 5 {∅, {∅}} ∈ V
1312rabex 4108 . . . 4 {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ∈ V
14 sseq1 3151 . . . . . 6 (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (𝑥 ⊆ ω ↔ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ⊆ ω))
15 eleq2 2221 . . . . . . 7 (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (𝑦𝑥𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}))
1615exbidv 1805 . . . . . 6 (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (∃𝑦 𝑦𝑥 ↔ ∃𝑦 𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}))
1714, 16anbi12d 465 . . . . 5 (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → ((𝑥 ⊆ ω ∧ ∃𝑦 𝑦𝑥) ↔ ({𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ⊆ ω ∧ ∃𝑦 𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))})))
18 eleq2 2221 . . . . . . . . . 10 (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (𝑧𝑥𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}))
1918notbid 657 . . . . . . . . 9 (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (¬ 𝑧𝑥 ↔ ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}))
2019imbi2d 229 . . . . . . . 8 (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → ((𝑧𝑦 → ¬ 𝑧𝑥) ↔ (𝑧𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))})))
2120albidv 1804 . . . . . . 7 (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥) ↔ ∀𝑧(𝑧𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))})))
2215, 21anbi12d 465 . . . . . 6 (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → ((𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥)) ↔ (𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}))))
2322exbidv 1805 . . . . 5 (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (∃𝑦(𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥)) ↔ ∃𝑦(𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}))))
2417, 23imbi12d 233 . . . 4 (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (((𝑥 ⊆ ω ∧ ∃𝑦 𝑦𝑥) → ∃𝑦(𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥))) ↔ (({𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ⊆ ω ∧ ∃𝑦 𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}) → ∃𝑦(𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))})))))
25 nnregexmid.1 . . . 4 ((𝑥 ⊆ ω ∧ ∃𝑦 𝑦𝑥) → ∃𝑦(𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥)))
2613, 24, 25vtocl 2766 . . 3 (({𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ⊆ ω ∧ ∃𝑦 𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}) → ∃𝑦(𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))})))
279, 11, 26mp2an 423 . 2 𝑦(𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}))
2810regexmidlem1 4492 . 2 (∃𝑦(𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))})) → (𝜑 ∨ ¬ 𝜑))
2927, 28ax-mp 5 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 698  wal 1333   = wceq 1335  wex 1472  wcel 2128  {crab 2439  wss 3102  c0 3394  {csn 3560  {cpr 3561  suc csuc 4325  ωcom 4549
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-uni 3773  df-int 3808  df-suc 4331  df-iom 4550
This theorem is referenced by: (None)
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