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Theorem reg2exmidlema 4452
Description: Lemma for reg2exmid 4454. 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 519 . . . . . . 7 (((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) ∧ 𝑢 = {∅}) → ∀𝑣𝐴 𝑢𝑣)
2 sseq1 3120 . . . . . . . . 9 (𝑢 = {∅} → (𝑢𝑣 ↔ {∅} ⊆ 𝑣))
32ralbidv 2437 . . . . . . . 8 (𝑢 = {∅} → (∀𝑣𝐴 𝑢𝑣 ↔ ∀𝑣𝐴 {∅} ⊆ 𝑣))
43adantl 275 . . . . . . 7 (((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) ∧ 𝑢 = {∅}) → (∀𝑣𝐴 𝑢𝑣 ↔ ∀𝑣𝐴 {∅} ⊆ 𝑣))
51, 4mpbid 146 . . . . . 6 (((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) ∧ 𝑢 = {∅}) → ∀𝑣𝐴 {∅} ⊆ 𝑣)
6 0ex 4058 . . . . . . . 8 ∅ ∈ V
76snss 3652 . . . . . . 7 (∅ ∈ 𝑣 ↔ {∅} ⊆ 𝑣)
87ralbii 2441 . . . . . 6 (∀𝑣𝐴 ∅ ∈ 𝑣 ↔ ∀𝑣𝐴 {∅} ⊆ 𝑣)
95, 8sylibr 133 . . . . 5 (((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) ∧ 𝑢 = {∅}) → ∀𝑣𝐴 ∅ ∈ 𝑣)
10 noel 3367 . . . . . 6 ¬ ∅ ∈ ∅
11 eqid 2139 . . . . . . . . . . . 12 ∅ = ∅
1211jctl 312 . . . . . . . . . . 11 (𝜑 → (∅ = ∅ ∧ 𝜑))
1312olcd 723 . . . . . . . . . 10 (𝜑 → (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑)))
146prid1 3632 . . . . . . . . . 10 ∅ ∈ {∅, {∅}}
1513, 14jctil 310 . . . . . . . . 9 (𝜑 → (∅ ∈ {∅, {∅}} ∧ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑))))
16 eqeq1 2146 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑥 = {∅} ↔ ∅ = {∅}))
17 eqeq1 2146 . . . . . . . . . . . 12 (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
1817anbi1d 460 . . . . . . . . . . 11 (𝑥 = ∅ → ((𝑥 = ∅ ∧ 𝜑) ↔ (∅ = ∅ ∧ 𝜑)))
1916, 18orbi12d 782 . . . . . . . . . 10 (𝑥 = ∅ → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑))))
20 regexmidlemm.a . . . . . . . . . 10 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
2119, 20elrab2 2843 . . . . . . . . 9 (∅ ∈ 𝐴 ↔ (∅ ∈ {∅, {∅}} ∧ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑))))
2215, 21sylibr 133 . . . . . . . 8 (𝜑 → ∅ ∈ 𝐴)
23 eleq2 2203 . . . . . . . . 9 (𝑣 = ∅ → (∅ ∈ 𝑣 ↔ ∅ ∈ ∅))
2423rspcv 2785 . . . . . . . 8 (∅ ∈ 𝐴 → (∀𝑣𝐴 ∅ ∈ 𝑣 → ∅ ∈ ∅))
2522, 24syl 14 . . . . . . 7 (𝜑 → (∀𝑣𝐴 ∅ ∈ 𝑣 → ∅ ∈ ∅))
2625com12 30 . . . . . 6 (∀𝑣𝐴 ∅ ∈ 𝑣 → (𝜑 → ∅ ∈ ∅))
2710, 26mtoi 653 . . . . 5 (∀𝑣𝐴 ∅ ∈ 𝑣 → ¬ 𝜑)
289, 27syl 14 . . . 4 (((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) ∧ 𝑢 = {∅}) → ¬ 𝜑)
2928olcd 723 . . 3 (((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) ∧ 𝑢 = {∅}) → (𝜑 ∨ ¬ 𝜑))
30 simprr 521 . . . 4 (((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) ∧ (𝑢 = ∅ ∧ 𝜑)) → 𝜑)
3130orcd 722 . . 3 (((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) ∧ (𝑢 = ∅ ∧ 𝜑)) → (𝜑 ∨ ¬ 𝜑))
32 eqeq1 2146 . . . . . . 7 (𝑥 = 𝑢 → (𝑥 = {∅} ↔ 𝑢 = {∅}))
33 eqeq1 2146 . . . . . . . 8 (𝑥 = 𝑢 → (𝑥 = ∅ ↔ 𝑢 = ∅))
3433anbi1d 460 . . . . . . 7 (𝑥 = 𝑢 → ((𝑥 = ∅ ∧ 𝜑) ↔ (𝑢 = ∅ ∧ 𝜑)))
3532, 34orbi12d 782 . . . . . 6 (𝑥 = 𝑢 → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))))
3635, 20elrab2 2843 . . . . 5 (𝑢𝐴 ↔ (𝑢 ∈ {∅, {∅}} ∧ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))))
3736simprbi 273 . . . 4 (𝑢𝐴 → (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑)))
3837adantr 274 . . 3 ((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) → (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑)))
3929, 31, 38mpjaodan 787 . 2 ((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) → (𝜑 ∨ ¬ 𝜑))
4039rexlimiva 2544 1 (∃𝑢𝐴𝑣𝐴 𝑢𝑣 → (𝜑 ∨ ¬ 𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 697   = wceq 1331  wcel 1480  wral 2416  wrex 2417  {crab 2420  wss 3071  c0 3363  {csn 3527  {cpr 3528
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-nul 4057
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-sn 3533  df-pr 3534
This theorem is referenced by:  reg2exmid  4454  reg3exmid  4497
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