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Theorem reg2exmidlema 4286
 Description: Lemma for reg2exmid 4288. 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 490 . . . . . . 7 (((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) ∧ 𝑢 = {∅}) → ∀𝑣𝐴 𝑢𝑣)
2 sseq1 2993 . . . . . . . . 9 (𝑢 = {∅} → (𝑢𝑣 ↔ {∅} ⊆ 𝑣))
32ralbidv 2343 . . . . . . . 8 (𝑢 = {∅} → (∀𝑣𝐴 𝑢𝑣 ↔ ∀𝑣𝐴 {∅} ⊆ 𝑣))
43adantl 266 . . . . . . 7 (((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) ∧ 𝑢 = {∅}) → (∀𝑣𝐴 𝑢𝑣 ↔ ∀𝑣𝐴 {∅} ⊆ 𝑣))
51, 4mpbid 139 . . . . . 6 (((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) ∧ 𝑢 = {∅}) → ∀𝑣𝐴 {∅} ⊆ 𝑣)
6 0ex 3911 . . . . . . . 8 ∅ ∈ V
76snss 3521 . . . . . . 7 (∅ ∈ 𝑣 ↔ {∅} ⊆ 𝑣)
87ralbii 2347 . . . . . 6 (∀𝑣𝐴 ∅ ∈ 𝑣 ↔ ∀𝑣𝐴 {∅} ⊆ 𝑣)
95, 8sylibr 141 . . . . 5 (((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) ∧ 𝑢 = {∅}) → ∀𝑣𝐴 ∅ ∈ 𝑣)
10 noel 3255 . . . . . 6 ¬ ∅ ∈ ∅
11 eqid 2056 . . . . . . . . . . . 12 ∅ = ∅
1211jctl 301 . . . . . . . . . . 11 (𝜑 → (∅ = ∅ ∧ 𝜑))
1312olcd 663 . . . . . . . . . 10 (𝜑 → (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑)))
146prid1 3503 . . . . . . . . . 10 ∅ ∈ {∅, {∅}}
1513, 14jctil 299 . . . . . . . . 9 (𝜑 → (∅ ∈ {∅, {∅}} ∧ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑))))
16 eqeq1 2062 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑥 = {∅} ↔ ∅ = {∅}))
17 eqeq1 2062 . . . . . . . . . . . 12 (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
1817anbi1d 446 . . . . . . . . . . 11 (𝑥 = ∅ → ((𝑥 = ∅ ∧ 𝜑) ↔ (∅ = ∅ ∧ 𝜑)))
1916, 18orbi12d 717 . . . . . . . . . 10 (𝑥 = ∅ → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑))))
20 regexmidlemm.a . . . . . . . . . 10 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
2119, 20elrab2 2722 . . . . . . . . 9 (∅ ∈ 𝐴 ↔ (∅ ∈ {∅, {∅}} ∧ (∅ = {∅} ∨ (∅ = ∅ ∧ 𝜑))))
2215, 21sylibr 141 . . . . . . . 8 (𝜑 → ∅ ∈ 𝐴)
23 eleq2 2117 . . . . . . . . 9 (𝑣 = ∅ → (∅ ∈ 𝑣 ↔ ∅ ∈ ∅))
2423rspcv 2669 . . . . . . . 8 (∅ ∈ 𝐴 → (∀𝑣𝐴 ∅ ∈ 𝑣 → ∅ ∈ ∅))
2522, 24syl 14 . . . . . . 7 (𝜑 → (∀𝑣𝐴 ∅ ∈ 𝑣 → ∅ ∈ ∅))
2625com12 30 . . . . . 6 (∀𝑣𝐴 ∅ ∈ 𝑣 → (𝜑 → ∅ ∈ ∅))
2710, 26mtoi 600 . . . . 5 (∀𝑣𝐴 ∅ ∈ 𝑣 → ¬ 𝜑)
289, 27syl 14 . . . 4 (((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) ∧ 𝑢 = {∅}) → ¬ 𝜑)
2928olcd 663 . . 3 (((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) ∧ 𝑢 = {∅}) → (𝜑 ∨ ¬ 𝜑))
30 simprr 492 . . . 4 (((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) ∧ (𝑢 = ∅ ∧ 𝜑)) → 𝜑)
3130orcd 662 . . 3 (((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) ∧ (𝑢 = ∅ ∧ 𝜑)) → (𝜑 ∨ ¬ 𝜑))
32 eqeq1 2062 . . . . . . 7 (𝑥 = 𝑢 → (𝑥 = {∅} ↔ 𝑢 = {∅}))
33 eqeq1 2062 . . . . . . . 8 (𝑥 = 𝑢 → (𝑥 = ∅ ↔ 𝑢 = ∅))
3433anbi1d 446 . . . . . . 7 (𝑥 = 𝑢 → ((𝑥 = ∅ ∧ 𝜑) ↔ (𝑢 = ∅ ∧ 𝜑)))
3532, 34orbi12d 717 . . . . . 6 (𝑥 = 𝑢 → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))))
3635, 20elrab2 2722 . . . . 5 (𝑢𝐴 ↔ (𝑢 ∈ {∅, {∅}} ∧ (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑))))
3736simprbi 264 . . . 4 (𝑢𝐴 → (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑)))
3837adantr 265 . . 3 ((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) → (𝑢 = {∅} ∨ (𝑢 = ∅ ∧ 𝜑)))
3929, 31, 38mpjaodan 722 . 2 ((𝑢𝐴 ∧ ∀𝑣𝐴 𝑢𝑣) → (𝜑 ∨ ¬ 𝜑))
4039rexlimiva 2445 1 (∃𝑢𝐴𝑣𝐴 𝑢𝑣 → (𝜑 ∨ ¬ 𝜑))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 101   ↔ wb 102   ∨ wo 639   = wceq 1259   ∈ wcel 1409  ∀wral 2323  ∃wrex 2324  {crab 2327   ⊆ wss 2944  ∅c0 3251  {csn 3402  {cpr 3403 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-nul 3910 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-sn 3408  df-pr 3409 This theorem is referenced by:  reg2exmid  4288  reg3exmid  4331
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