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Theorem reg3exmid 4554
Description: If any inhabited set satisfying df-wetr 4309 for E has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Oct-2021.)
Hypothesis
Ref Expression
reg3exmid.1 (( E We 𝑧 ∧ ∃𝑤 𝑤𝑧) → ∃𝑥𝑧𝑦𝑧 𝑥𝑦)
Assertion
Ref Expression
reg3exmid (𝜑 ∨ ¬ 𝜑)
Distinct variable groups:   𝜑,𝑤,𝑧   𝜑,𝑥,𝑦,𝑧

Proof of Theorem reg3exmid
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 eqid 2164 . . 3 {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} = {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}
21regexmidlemm 4506 . 2 𝑤 𝑤 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}
31reg3exmidlemwe 4553 . . 3 E We {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}
4 pp0ex 4165 . . . . 5 {∅, {∅}} ∈ V
54rabex 4123 . . . 4 {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} ∈ V
6 weeq2 4332 . . . . . 6 (𝑧 = {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} → ( E We 𝑧 ↔ E We {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}))
7 eleq2 2228 . . . . . . 7 (𝑧 = {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} → (𝑤𝑧𝑤 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}))
87exbidv 1812 . . . . . 6 (𝑧 = {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} → (∃𝑤 𝑤𝑧 ↔ ∃𝑤 𝑤 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}))
96, 8anbi12d 465 . . . . 5 (𝑧 = {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} → (( E We 𝑧 ∧ ∃𝑤 𝑤𝑧) ↔ ( E We {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} ∧ ∃𝑤 𝑤 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))})))
10 raleq 2659 . . . . . 6 (𝑧 = {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} → (∀𝑦𝑧 𝑥𝑦 ↔ ∀𝑦 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}𝑥𝑦))
1110rexeqbi1dv 2668 . . . . 5 (𝑧 = {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} → (∃𝑥𝑧𝑦𝑧 𝑥𝑦 ↔ ∃𝑥 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}∀𝑦 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}𝑥𝑦))
129, 11imbi12d 233 . . . 4 (𝑧 = {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} → ((( E We 𝑧 ∧ ∃𝑤 𝑤𝑧) → ∃𝑥𝑧𝑦𝑧 𝑥𝑦) ↔ (( E We {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} ∧ ∃𝑤 𝑤 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}) → ∃𝑥 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}∀𝑦 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}𝑥𝑦)))
13 reg3exmid.1 . . . 4 (( E We 𝑧 ∧ ∃𝑤 𝑤𝑧) → ∃𝑥𝑧𝑦𝑧 𝑥𝑦)
145, 12, 13vtocl 2778 . . 3 (( E We {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} ∧ ∃𝑤 𝑤 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}) → ∃𝑥 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}∀𝑦 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}𝑥𝑦)
153, 14mpan 421 . 2 (∃𝑤 𝑤 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))} → ∃𝑥 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}∀𝑦 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}𝑥𝑦)
161reg2exmidlema 4508 . 2 (∃𝑥 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}∀𝑦 ∈ {𝑢 ∈ {∅, {∅}} ∣ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))}𝑥𝑦 → (𝜑 ∨ ¬ 𝜑))
172, 15, 16mp2b 8 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 698   = wceq 1342  wex 1479  wcel 2135  wral 2442  wrex 2443  {crab 2446  wss 3114  c0 3407  {csn 3573  {cpr 3574   E cep 4262   We wwe 4305
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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4097  ax-nul 4105  ax-pow 4150  ax-pr 4184  ax-setind 4511
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2726  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3408  df-pw 3558  df-sn 3579  df-pr 3580  df-op 3582  df-br 3980  df-opab 4041  df-eprel 4264  df-frfor 4306  df-frind 4307  df-wetr 4309
This theorem is referenced by: (None)
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