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Theorem reg2exmidlema 4313
Description: Lemma for reg2exmid 4315. If 𝐴 has a minimal element (expressed by ), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.)
Hypothesis
Ref Expression
regexmidlemm.a 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
Assertion
Ref Expression
reg2exmidlema (∃𝑢𝐴𝑣𝐴 𝑢𝑣 → (𝜑 ∨ ¬ 𝜑))
Distinct variable groups:   𝜑,𝑥   𝑣,𝐴   𝜑,𝑢,𝑥   𝑣,𝑢
Allowed substitution hints:   𝜑(𝑣)   𝐴(𝑥,𝑢)

Proof of Theorem reg2exmidlema
StepHypRef Expression
1 simplr 497 . . . . . . 7 (((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) ∧ 𝑢 = {∅}) → ∀𝑣𝐴 𝑢𝑣)
2 sseq1 3031 . . . . . . . . 9 (𝑢 = {∅} → (𝑢𝑣 ↔ {∅} ⊆ 𝑣))
32ralbidv 2374 . . . . . . . 8 (𝑢 = {∅} → (∀𝑣𝐴 𝑢𝑣 ↔ ∀𝑣𝐴 {∅} ⊆ 𝑣))
43adantl 271 . . . . . . 7 (((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) ∧ 𝑢 = {∅}) → (∀𝑣𝐴 𝑢𝑣 ↔ ∀𝑣𝐴 {∅} ⊆ 𝑣))
51, 4mpbid 145 . . . . . 6 (((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) ∧ 𝑢 = {∅}) → ∀𝑣𝐴 {∅} ⊆ 𝑣)
6 0ex 3931 . . . . . . . 8 ∅ ∈ V
76snss 3540 . . . . . . 7 (∅ ∈ 𝑣 ↔ {∅} ⊆ 𝑣)
87ralbii 2378 . . . . . 6 (∀𝑣𝐴 ∅ ∈ 𝑣 ↔ ∀𝑣𝐴 {∅} ⊆ 𝑣)
95, 8sylibr 132 . . . . 5 (((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) ∧ 𝑢 = {∅}) → ∀𝑣𝐴 ∅ ∈ 𝑣)
10 noel 3273 . . . . . 6 ¬ ∅ ∈ ∅
11 eqid 2083 . . . . . . . . . . . 12 ∅ = ∅
1211jctl 307 . . . . . . . . . . 11 (𝜑 → (∅ = ∅ ∧ 𝜑))
1312olcd 686 . . . . . . . . . 10 (𝜑 → (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑)))
146prid1 3522 . . . . . . . . . 10 ∅ ∈ {∅, {∅}}
1513, 14jctil 305 . . . . . . . . 9 (𝜑 → (∅ ∈ {∅, {∅}} ∧ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑))))
16 eqeq1 2089 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑥 = {∅} ↔ ∅ = {∅}))
17 eqeq1 2089 . . . . . . . . . . . 12 (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
1817anbi1d 453 . . . . . . . . . . 11 (𝑥 = ∅ → ((𝑥 = ∅ ∧ 𝜑) ↔ (∅ = ∅ ∧ 𝜑)))
1916, 18orbi12d 740 . . . . . . . . . 10 (𝑥 = ∅ → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑))))
20 regexmidlemm.a . . . . . . . . . 10 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
2119, 20elrab2 2762 . . . . . . . . 9 (∅ ∈ 𝐴 ↔ (∅ ∈ {∅, {∅}} ∧ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑))))
2215, 21sylibr 132 . . . . . . . 8 (𝜑 → ∅ ∈ 𝐴)
23 eleq2 2146 . . . . . . . . 9 (𝑣 = ∅ → (∅ ∈ 𝑣 ↔ ∅ ∈ ∅))
2423rspcv 2708 . . . . . . . 8 (∅ ∈ 𝐴 → (∀𝑣𝐴 ∅ ∈ 𝑣 → ∅ ∈ ∅))
2522, 24syl 14 . . . . . . 7 (𝜑 → (∀𝑣𝐴 ∅ ∈ 𝑣 → ∅ ∈ ∅))
2625com12 30 . . . . . 6 (∀𝑣𝐴 ∅ ∈ 𝑣 → (𝜑 → ∅ ∈ ∅))
2710, 26mtoi 623 . . . . 5 (∀𝑣𝐴 ∅ ∈ 𝑣 → ¬ 𝜑)
289, 27syl 14 . . . 4 (((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) ∧ 𝑢 = {∅}) → ¬ 𝜑)
2928olcd 686 . . 3 (((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) ∧ 𝑢 = {∅}) → (𝜑 ∨ ¬ 𝜑))
30 simprr 499 . . . 4 (((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) ∧ (𝑢 = ∅ ∧ 𝜑)) → 𝜑)
3130orcd 685 . . 3 (((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) ∧ (𝑢 = ∅ ∧ 𝜑)) → (𝜑 ∨ ¬ 𝜑))
32 eqeq1 2089 . . . . . . 7 (𝑥 = 𝑢 → (𝑥 = {∅} ↔ 𝑢 = {∅}))
33 eqeq1 2089 . . . . . . . 8 (𝑥 = 𝑢 → (𝑥 = ∅ ↔ 𝑢 = ∅))
3433anbi1d 453 . . . . . . 7 (𝑥 = 𝑢 → ((𝑥 = ∅ ∧ 𝜑) ↔ (𝑢 = ∅ ∧ 𝜑)))
3532, 34orbi12d 740 . . . . . 6 (𝑥 = 𝑢 → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))))
3635, 20elrab2 2762 . . . . 5 (𝑢𝐴 ↔ (𝑢 ∈ {∅, {∅}} ∧ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))))
3736simprbi 269 . . . 4 (𝑢𝐴 → (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑)))
3837adantr 270 . . 3 ((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) → (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑)))
3929, 31, 38mpjaodan 745 . 2 ((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) → (𝜑 ∨ ¬ 𝜑))
4039rexlimiva 2478 1 (∃𝑢𝐴𝑣𝐴 𝑢𝑣 → (𝜑 ∨ ¬ 𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662   = wceq 1285  wcel 1434  wral 2353  wrex 2354  {crab 2357  wss 2984  c0 3269  {csn 3422  {cpr 3423
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-nul 3930
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-sn 3428  df-pr 3429
This theorem is referenced by:  reg2exmid  4315  reg3exmid  4358
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