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Theorem regexmidlem1 4526
Description: Lemma for regexmid 4528. 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 2182 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 = {∅} ↔ 𝑦 = {∅}))
2 eqeq1 2182 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
32anbi1d 465 . . . . . . 7 (𝑥 = 𝑦 → ((𝑥 = ∅ ∧ 𝜑) ↔ (𝑦 = ∅ ∧ 𝜑)))
41, 3orbi12d 793 . . . . . 6 (𝑥 = 𝑦 → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ (𝑦 = {∅} ∨ (𝑦 = ∅ ∧ 𝜑))))
5 regexmidlemm.a . . . . . 6 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
64, 5elrab2 2894 . . . . 5 (𝑦𝐴 ↔ (𝑦 ∈ {∅, {∅}} ∧ (𝑦 = {∅} ∨ (𝑦 = ∅ ∧ 𝜑))))
76simprbi 275 . . . 4 (𝑦𝐴 → (𝑦 = {∅} ∨ (𝑦 = ∅ ∧ 𝜑)))
8 0ex 4125 . . . . . . . . 9 ∅ ∈ V
98snid 3620 . . . . . . . 8 ∅ ∈ {∅}
10 eleq2 2239 . . . . . . . 8 (𝑦 = {∅} → (∅ ∈ 𝑦 ↔ ∅ ∈ {∅}))
119, 10mpbiri 168 . . . . . . 7 (𝑦 = {∅} → ∅ ∈ 𝑦)
12 eleq1 2238 . . . . . . . . 9 (𝑧 = ∅ → (𝑧𝑦 ↔ ∅ ∈ 𝑦))
13 eleq1 2238 . . . . . . . . . 10 (𝑧 = ∅ → (𝑧𝐴 ↔ ∅ ∈ 𝐴))
1413notbid 667 . . . . . . . . 9 (𝑧 = ∅ → (¬ 𝑧𝐴 ↔ ¬ ∅ ∈ 𝐴))
1512, 14imbi12d 234 . . . . . . . 8 (𝑧 = ∅ → ((𝑧𝑦 → ¬ 𝑧𝐴) ↔ (∅ ∈ 𝑦 → ¬ ∅ ∈ 𝐴)))
168, 15spcv 2829 . . . . . . 7 (∀𝑧(𝑧𝑦 → ¬ 𝑧𝐴) → (∅ ∈ 𝑦 → ¬ ∅ ∈ 𝐴))
1711, 16syl5com 29 . . . . . 6 (𝑦 = {∅} → (∀𝑧(𝑧𝑦 → ¬ 𝑧𝐴) → ¬ ∅ ∈ 𝐴))
188prid1 3695 . . . . . . . . . 10 ∅ ∈ {∅, {∅}}
19 eqeq1 2182 . . . . . . . . . . . 12 (𝑥 = ∅ → (𝑥 = {∅} ↔ ∅ = {∅}))
20 eqeq1 2182 . . . . . . . . . . . . 13 (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
2120anbi1d 465 . . . . . . . . . . . 12 (𝑥 = ∅ → ((𝑥 = ∅ ∧ 𝜑) ↔ (∅ = ∅ ∧ 𝜑)))
2219, 21orbi12d 793 . . . . . . . . . . 11 (𝑥 = ∅ → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑))))
2322, 5elrab2 2894 . . . . . . . . . 10 (∅ ∈ 𝐴 ↔ (∅ ∈ {∅, {∅}} ∧ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑))))
2418, 23mpbiran 940 . . . . . . . . 9 (∅ ∈ 𝐴 ↔ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑)))
25 pm2.46 739 . . . . . . . . 9 (¬ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑)) → ¬ (∅ = ∅ ∧ 𝜑))
2624, 25sylnbi 678 . . . . . . . 8 (¬ ∅ ∈ 𝐴 → ¬ (∅ = ∅ ∧ 𝜑))
27 eqid 2175 . . . . . . . . 9 ∅ = ∅
2827biantrur 303 . . . . . . . 8 (𝜑 ↔ (∅ = ∅ ∧ 𝜑))
2926, 28sylnibr 677 . . . . . . 7 (¬ ∅ ∈ 𝐴 → ¬ 𝜑)
3029olcd 734 . . . . . 6 (¬ ∅ ∈ 𝐴 → (𝜑 ∨ ¬ 𝜑))
3117, 30syl6 33 . . . . 5 (𝑦 = {∅} → (∀𝑧(𝑧𝑦 → ¬ 𝑧𝐴) → (𝜑 ∨ ¬ 𝜑)))
32 orc 712 . . . . . . 7 (𝜑 → (𝜑 ∨ ¬ 𝜑))
3332adantl 277 . . . . . 6 ((𝑦 = ∅ ∧ 𝜑) → (𝜑 ∨ ¬ 𝜑))
3433a1d 22 . . . . 5 ((𝑦 = ∅ ∧ 𝜑) → (∀𝑧(𝑧𝑦 → ¬ 𝑧𝐴) → (𝜑 ∨ ¬ 𝜑)))
3531, 34jaoi 716 . . . 4 ((𝑦 = {∅} ∨ (𝑦 = ∅ ∧ 𝜑)) → (∀𝑧(𝑧𝑦 → ¬ 𝑧𝐴) → (𝜑 ∨ ¬ 𝜑)))
367, 35syl 14 . . 3 (𝑦𝐴 → (∀𝑧(𝑧𝑦 → ¬ 𝑧𝐴) → (𝜑 ∨ ¬ 𝜑)))
3736imp 124 . 2 ((𝑦𝐴 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝐴)) → (𝜑 ∨ ¬ 𝜑))
3837exlimiv 1596 1 (∃𝑦(𝑦𝐴 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝐴)) → (𝜑 ∨ ¬ 𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 708  wal 1351   = wceq 1353  wex 1490  wcel 2146  {crab 2457  c0 3420  {csn 3589  {cpr 3590
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157  ax-nul 4124
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rab 2462  df-v 2737  df-dif 3129  df-un 3131  df-nul 3421  df-sn 3595  df-pr 3596
This theorem is referenced by:  regexmid  4528  nnregexmid  4614
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