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Theorem regexmid 4379
Description: The axiom of foundation implies excluded middle.

By foundation (or regularity), we mean the principle that every inhabited set has an element which is minimal (when arranged by ). The statement of foundation here is taken from Metamath Proof Explorer's ax-reg, and is identical (modulo one unnecessary quantifier) to the statement of foundation in Theorem "Foundation implies instances of EM" of [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic".

For this reason, IZF does not adopt foundation as an axiom and instead replaces it with ax-setind 4381. (Contributed by Jim Kingdon, 3-Sep-2019.)

Hypothesis
Ref Expression
regexmid.1 (∃𝑦 𝑦𝑥 → ∃𝑦(𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥)))
Assertion
Ref Expression
regexmid (𝜑 ∨ ¬ 𝜑)
Distinct variable group:   𝜑,𝑥,𝑦,𝑧

Proof of Theorem regexmid
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 eqid 2095 . . 3 {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}
21regexmidlemm 4376 . 2 𝑦 𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}
3 pp0ex 4045 . . . 4 {∅, {∅}} ∈ V
43rabex 4004 . . 3 {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ∈ V
5 eleq2 2158 . . . . 5 (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (𝑦𝑥𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}))
65exbidv 1760 . . . 4 (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (∃𝑦 𝑦𝑥 ↔ ∃𝑦 𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}))
7 eleq2 2158 . . . . . . . . 9 (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (𝑧𝑥𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}))
87notbid 630 . . . . . . . 8 (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (¬ 𝑧𝑥 ↔ ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}))
98imbi2d 229 . . . . . . 7 (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → ((𝑧𝑦 → ¬ 𝑧𝑥) ↔ (𝑧𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))})))
109albidv 1759 . . . . . 6 (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥) ↔ ∀𝑧(𝑧𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))})))
115, 10anbi12d 458 . . . . 5 (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → ((𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥)) ↔ (𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}))))
1211exbidv 1760 . . . 4 (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → (∃𝑦(𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥)) ↔ ∃𝑦(𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))}))))
136, 12imbi12d 233 . . 3 (𝑥 = {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → ((∃𝑦 𝑦𝑥 → ∃𝑦(𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥))) ↔ (∃𝑦 𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → ∃𝑦(𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))})))))
14 regexmid.1 . . 3 (∃𝑦 𝑦𝑥 → ∃𝑦(𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥)))
154, 13, 14vtocl 2687 . 2 (∃𝑦 𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} → ∃𝑦(𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))})))
161regexmidlem1 4377 . 2 (∃𝑦(𝑦 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))} ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧 ∈ {𝑤 ∈ {∅, {∅}} ∣ (𝑤 = {∅} ∨ (𝑤 = ∅ ∧ 𝜑))})) → (𝜑 ∨ ¬ 𝜑))
172, 15, 16mp2b 8 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 667  wal 1294   = wceq 1296  wex 1433  wcel 1445  {crab 2374  c0 3302  {csn 3466  {cpr 3467
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-nul 3986  ax-pow 4030
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-rab 2379  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473
This theorem is referenced by: (None)
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