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Theorem regexmidlem1 4285
Description: Lemma for regexmid 4287. If 𝐴 has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2019.)
Hypothesis
Ref Expression
regexmidlemm.a 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
Assertion
Ref Expression
regexmidlem1 (∃𝑦(𝑦𝐴 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝐴)) → (𝜑 ∨ ¬ 𝜑))
Distinct variable groups:   𝑦,𝐴,𝑧   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑧)   𝐴(𝑥)

Proof of Theorem regexmidlem1
StepHypRef Expression
1 eqeq1 2062 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 = {∅} ↔ 𝑦 = {∅}))
2 eqeq1 2062 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
32anbi1d 446 . . . . . . 7 (𝑥 = 𝑦 → ((𝑥 = ∅ ∧ 𝜑) ↔ (𝑦 = ∅ ∧ 𝜑)))
41, 3orbi12d 717 . . . . . 6 (𝑥 = 𝑦 → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ (𝑦 = {∅} ∨ (𝑦 = ∅ ∧ 𝜑))))
5 regexmidlemm.a . . . . . 6 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
64, 5elrab2 2722 . . . . 5 (𝑦𝐴 ↔ (𝑦 ∈ {∅, {∅}} ∧ (𝑦 = {∅} ∨ (𝑦 = ∅ ∧ 𝜑))))
76simprbi 264 . . . 4 (𝑦𝐴 → (𝑦 = {∅} ∨ (𝑦 = ∅ ∧ 𝜑)))
8 0ex 3911 . . . . . . . . 9 ∅ ∈ V
98snid 3429 . . . . . . . 8 ∅ ∈ {∅}
10 eleq2 2117 . . . . . . . 8 (𝑦 = {∅} → (∅ ∈ 𝑦 ↔ ∅ ∈ {∅}))
119, 10mpbiri 161 . . . . . . 7 (𝑦 = {∅} → ∅ ∈ 𝑦)
12 eleq1 2116 . . . . . . . . 9 (𝑧 = ∅ → (𝑧𝑦 ↔ ∅ ∈ 𝑦))
13 eleq1 2116 . . . . . . . . . 10 (𝑧 = ∅ → (𝑧𝐴 ↔ ∅ ∈ 𝐴))
1413notbid 602 . . . . . . . . 9 (𝑧 = ∅ → (¬ 𝑧𝐴 ↔ ¬ ∅ ∈ 𝐴))
1512, 14imbi12d 227 . . . . . . . 8 (𝑧 = ∅ → ((𝑧𝑦 → ¬ 𝑧𝐴) ↔ (∅ ∈ 𝑦 → ¬ ∅ ∈ 𝐴)))
168, 15spcv 2663 . . . . . . 7 (∀𝑧(𝑧𝑦 → ¬ 𝑧𝐴) → (∅ ∈ 𝑦 → ¬ ∅ ∈ 𝐴))
1711, 16syl5com 29 . . . . . 6 (𝑦 = {∅} → (∀𝑧(𝑧𝑦 → ¬ 𝑧𝐴) → ¬ ∅ ∈ 𝐴))
188prid1 3503 . . . . . . . . . 10 ∅ ∈ {∅, {∅}}
19 eqeq1 2062 . . . . . . . . . . . 12 (𝑥 = ∅ → (𝑥 = {∅} ↔ ∅ = {∅}))
20 eqeq1 2062 . . . . . . . . . . . . 13 (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
2120anbi1d 446 . . . . . . . . . . . 12 (𝑥 = ∅ → ((𝑥 = ∅ ∧ 𝜑) ↔ (∅ = ∅ ∧ 𝜑)))
2219, 21orbi12d 717 . . . . . . . . . . 11 (𝑥 = ∅ → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑))))
2322, 5elrab2 2722 . . . . . . . . . 10 (∅ ∈ 𝐴 ↔ (∅ ∈ {∅, {∅}} ∧ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑))))
2418, 23mpbiran 858 . . . . . . . . 9 (∅ ∈ 𝐴 ↔ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑)))
25 pm2.46 668 . . . . . . . . 9 (¬ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑)) → ¬ (∅ = ∅ ∧ 𝜑))
2624, 25sylnbi 613 . . . . . . . 8 (¬ ∅ ∈ 𝐴 → ¬ (∅ = ∅ ∧ 𝜑))
27 eqid 2056 . . . . . . . . 9 ∅ = ∅
2827biantrur 291 . . . . . . . 8 (𝜑 ↔ (∅ = ∅ ∧ 𝜑))
2926, 28sylnibr 612 . . . . . . 7 (¬ ∅ ∈ 𝐴 → ¬ 𝜑)
3029olcd 663 . . . . . 6 (¬ ∅ ∈ 𝐴 → (𝜑 ∨ ¬ 𝜑))
3117, 30syl6 33 . . . . 5 (𝑦 = {∅} → (∀𝑧(𝑧𝑦 → ¬ 𝑧𝐴) → (𝜑 ∨ ¬ 𝜑)))
32 orc 643 . . . . . . 7 (𝜑 → (𝜑 ∨ ¬ 𝜑))
3332adantl 266 . . . . . 6 ((𝑦 = ∅ ∧ 𝜑) → (𝜑 ∨ ¬ 𝜑))
3433a1d 22 . . . . 5 ((𝑦 = ∅ ∧ 𝜑) → (∀𝑧(𝑧𝑦 → ¬ 𝑧𝐴) → (𝜑 ∨ ¬ 𝜑)))
3531, 34jaoi 646 . . . 4 ((𝑦 = {∅} ∨ (𝑦 = ∅ ∧ 𝜑)) → (∀𝑧(𝑧𝑦 → ¬ 𝑧𝐴) → (𝜑 ∨ ¬ 𝜑)))
367, 35syl 14 . . 3 (𝑦𝐴 → (∀𝑧(𝑧𝑦 → ¬ 𝑧𝐴) → (𝜑 ∨ ¬ 𝜑)))
3736imp 119 . 2 ((𝑦𝐴 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝐴)) → (𝜑 ∨ ¬ 𝜑))
3837exlimiv 1505 1 (∃𝑦(𝑦𝐴 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝐴)) → (𝜑 ∨ ¬ 𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wo 639  wal 1257   = wceq 1259  wex 1397  wcel 1409  {crab 2327  c0 3251  {csn 3402  {cpr 3403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-nul 3910
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rab 2332  df-v 2576  df-dif 2947  df-un 2949  df-nul 3252  df-sn 3408  df-pr 3409
This theorem is referenced by:  regexmid  4287  nnregexmid  4369
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