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Theorem reg2exmidlema 4340
Description: Lemma for reg2exmid 4342. 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 3045 . . . . . . . . 9 (𝑢 = {∅} → (𝑢𝑣 ↔ {∅} ⊆ 𝑣))
32ralbidv 2380 . . . . . . . 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 3958 . . . . . . . 8 ∅ ∈ V
76snss 3561 . . . . . . 7 (∅ ∈ 𝑣 ↔ {∅} ⊆ 𝑣)
87ralbii 2384 . . . . . 6 (∀𝑣𝐴 ∅ ∈ 𝑣 ↔ ∀𝑣𝐴 {∅} ⊆ 𝑣)
95, 8sylibr 132 . . . . 5 (((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) ∧ 𝑢 = {∅}) → ∀𝑣𝐴 ∅ ∈ 𝑣)
10 noel 3288 . . . . . 6 ¬ ∅ ∈ ∅
11 eqid 2088 . . . . . . . . . . . 12 ∅ = ∅
1211jctl 307 . . . . . . . . . . 11 (𝜑 → (∅ = ∅ ∧ 𝜑))
1312olcd 688 . . . . . . . . . 10 (𝜑 → (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑)))
146prid1 3543 . . . . . . . . . 10 ∅ ∈ {∅, {∅}}
1513, 14jctil 305 . . . . . . . . 9 (𝜑 → (∅ ∈ {∅, {∅}} ∧ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑))))
16 eqeq1 2094 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑥 = {∅} ↔ ∅ = {∅}))
17 eqeq1 2094 . . . . . . . . . . . 12 (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
1817anbi1d 453 . . . . . . . . . . 11 (𝑥 = ∅ → ((𝑥 = ∅ ∧ 𝜑) ↔ (∅ = ∅ ∧ 𝜑)))
1916, 18orbi12d 742 . . . . . . . . . 10 (𝑥 = ∅ → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑))))
20 regexmidlemm.a . . . . . . . . . 10 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
2119, 20elrab2 2772 . . . . . . . . 9 (∅ ∈ 𝐴 ↔ (∅ ∈ {∅, {∅}} ∧ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑))))
2215, 21sylibr 132 . . . . . . . 8 (𝜑 → ∅ ∈ 𝐴)
23 eleq2 2151 . . . . . . . . 9 (𝑣 = ∅ → (∅ ∈ 𝑣 ↔ ∅ ∈ ∅))
2423rspcv 2718 . . . . . . . 8 (∅ ∈ 𝐴 → (∀𝑣𝐴 ∅ ∈ 𝑣 → ∅ ∈ ∅))
2522, 24syl 14 . . . . . . 7 (𝜑 → (∀𝑣𝐴 ∅ ∈ 𝑣 → ∅ ∈ ∅))
2625com12 30 . . . . . 6 (∀𝑣𝐴 ∅ ∈ 𝑣 → (𝜑 → ∅ ∈ ∅))
2710, 26mtoi 625 . . . . 5 (∀𝑣𝐴 ∅ ∈ 𝑣 → ¬ 𝜑)
289, 27syl 14 . . . 4 (((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) ∧ 𝑢 = {∅}) → ¬ 𝜑)
2928olcd 688 . . 3 (((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) ∧ 𝑢 = {∅}) → (𝜑 ∨ ¬ 𝜑))
30 simprr 499 . . . 4 (((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) ∧ (𝑢 = ∅ ∧ 𝜑)) → 𝜑)
3130orcd 687 . . 3 (((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) ∧ (𝑢 = ∅ ∧ 𝜑)) → (𝜑 ∨ ¬ 𝜑))
32 eqeq1 2094 . . . . . . 7 (𝑥 = 𝑢 → (𝑥 = {∅} ↔ 𝑢 = {∅}))
33 eqeq1 2094 . . . . . . . 8 (𝑥 = 𝑢 → (𝑥 = ∅ ↔ 𝑢 = ∅))
3433anbi1d 453 . . . . . . 7 (𝑥 = 𝑢 → ((𝑥 = ∅ ∧ 𝜑) ↔ (𝑢 = ∅ ∧ 𝜑)))
3532, 34orbi12d 742 . . . . . 6 (𝑥 = 𝑢 → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))))
3635, 20elrab2 2772 . . . . 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 747 . 2 ((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) → (𝜑 ∨ ¬ 𝜑))
4039rexlimiva 2484 1 (∃𝑢𝐴𝑣𝐴 𝑢𝑣 → (𝜑 ∨ ¬ 𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 664   = wceq 1289  wcel 1438  wral 2359  wrex 2360  {crab 2363  wss 2997  c0 3284  {csn 3441  {cpr 3442
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-nul 3957
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-sn 3447  df-pr 3448
This theorem is referenced by:  reg2exmid  4342  reg3exmid  4385
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