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Theorem reg3exmidlemwe 4596
Description: Lemma for reg3exmid 4597. Our counterexample 𝐴 satisfies We. (Contributed by Jim Kingdon, 3-Oct-2021.)
Hypothesis
Ref Expression
reg3exmidlemwe.a 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
Assertion
Ref Expression
reg3exmidlemwe E We 𝐴
Distinct variable group:   𝜑,𝑥
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem reg3exmidlemwe
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zfregfr 4591 . 2 E Fr 𝐴
2 epel 4310 . . . . . 6 (𝑎 E 𝑏𝑎𝑏)
3 epel 4310 . . . . . 6 (𝑏 E 𝑐𝑏𝑐)
42, 3anbi12i 460 . . . . 5 ((𝑎 E 𝑏𝑏 E 𝑐) ↔ (𝑎𝑏𝑏𝑐))
5 simpr 110 . . . . . 6 (((𝑎𝐴𝑏𝐴𝑐𝐴) ∧ (𝑎𝑏𝑏𝑐)) → (𝑎𝑏𝑏𝑐))
6 elirr 4558 . . . . . . . 8 ¬ {∅} ∈ {∅}
7 simprr 531 . . . . . . . . . 10 (((𝑎𝐴𝑏𝐴𝑐𝐴) ∧ (𝑎𝑏𝑏𝑐)) → 𝑏𝑐)
8 noel 3441 . . . . . . . . . . . . 13 ¬ 𝑎 ∈ ∅
9 eleq2 2253 . . . . . . . . . . . . 13 (𝑏 = ∅ → (𝑎𝑏𝑎 ∈ ∅))
108, 9mtbiri 676 . . . . . . . . . . . 12 (𝑏 = ∅ → ¬ 𝑎𝑏)
11 simprl 529 . . . . . . . . . . . 12 (((𝑎𝐴𝑏𝐴𝑐𝐴) ∧ (𝑎𝑏𝑏𝑐)) → 𝑎𝑏)
1210, 11nsyl3 627 . . . . . . . . . . 11 (((𝑎𝐴𝑏𝐴𝑐𝐴) ∧ (𝑎𝑏𝑏𝑐)) → ¬ 𝑏 = ∅)
13 elrabi 2905 . . . . . . . . . . . . . . . 16 (𝑏 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))} → 𝑏 ∈ {∅, {∅}})
14 reg3exmidlemwe.a . . . . . . . . . . . . . . . 16 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
1513, 14eleq2s 2284 . . . . . . . . . . . . . . 15 (𝑏𝐴𝑏 ∈ {∅, {∅}})
16 elpri 3630 . . . . . . . . . . . . . . 15 (𝑏 ∈ {∅, {∅}} → (𝑏 = ∅ ∨ 𝑏 = {∅}))
1715, 16syl 14 . . . . . . . . . . . . . 14 (𝑏𝐴 → (𝑏 = ∅ ∨ 𝑏 = {∅}))
1817orcomd 730 . . . . . . . . . . . . 13 (𝑏𝐴 → (𝑏 = {∅} ∨ 𝑏 = ∅))
19183ad2ant2 1021 . . . . . . . . . . . 12 ((𝑎𝐴𝑏𝐴𝑐𝐴) → (𝑏 = {∅} ∨ 𝑏 = ∅))
2019adantr 276 . . . . . . . . . . 11 (((𝑎𝐴𝑏𝐴𝑐𝐴) ∧ (𝑎𝑏𝑏𝑐)) → (𝑏 = {∅} ∨ 𝑏 = ∅))
2112, 20ecased 1360 . . . . . . . . . 10 (((𝑎𝐴𝑏𝐴𝑐𝐴) ∧ (𝑎𝑏𝑏𝑐)) → 𝑏 = {∅})
22 noel 3441 . . . . . . . . . . . . 13 ¬ 𝑏 ∈ ∅
23 eleq2 2253 . . . . . . . . . . . . 13 (𝑐 = ∅ → (𝑏𝑐𝑏 ∈ ∅))
2422, 23mtbiri 676 . . . . . . . . . . . 12 (𝑐 = ∅ → ¬ 𝑏𝑐)
2524, 7nsyl3 627 . . . . . . . . . . 11 (((𝑎𝐴𝑏𝐴𝑐𝐴) ∧ (𝑎𝑏𝑏𝑐)) → ¬ 𝑐 = ∅)
26 elrabi 2905 . . . . . . . . . . . . . . . 16 (𝑐 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))} → 𝑐 ∈ {∅, {∅}})
2726, 14eleq2s 2284 . . . . . . . . . . . . . . 15 (𝑐𝐴𝑐 ∈ {∅, {∅}})
28 vex 2755 . . . . . . . . . . . . . . . 16 𝑐 ∈ V
2928elpr 3628 . . . . . . . . . . . . . . 15 (𝑐 ∈ {∅, {∅}} ↔ (𝑐 = ∅ ∨ 𝑐 = {∅}))
3027, 29sylib 122 . . . . . . . . . . . . . 14 (𝑐𝐴 → (𝑐 = ∅ ∨ 𝑐 = {∅}))
3130orcomd 730 . . . . . . . . . . . . 13 (𝑐𝐴 → (𝑐 = {∅} ∨ 𝑐 = ∅))
32313ad2ant3 1022 . . . . . . . . . . . 12 ((𝑎𝐴𝑏𝐴𝑐𝐴) → (𝑐 = {∅} ∨ 𝑐 = ∅))
3332adantr 276 . . . . . . . . . . 11 (((𝑎𝐴𝑏𝐴𝑐𝐴) ∧ (𝑎𝑏𝑏𝑐)) → (𝑐 = {∅} ∨ 𝑐 = ∅))
3425, 33ecased 1360 . . . . . . . . . 10 (((𝑎𝐴𝑏𝐴𝑐𝐴) ∧ (𝑎𝑏𝑏𝑐)) → 𝑐 = {∅})
357, 21, 343eltr3d 2272 . . . . . . . . 9 (((𝑎𝐴𝑏𝐴𝑐𝐴) ∧ (𝑎𝑏𝑏𝑐)) → {∅} ∈ {∅})
3635ex 115 . . . . . . . 8 ((𝑎𝐴𝑏𝐴𝑐𝐴) → ((𝑎𝑏𝑏𝑐) → {∅} ∈ {∅}))
376, 36mtoi 665 . . . . . . 7 ((𝑎𝐴𝑏𝐴𝑐𝐴) → ¬ (𝑎𝑏𝑏𝑐))
3837adantr 276 . . . . . 6 (((𝑎𝐴𝑏𝐴𝑐𝐴) ∧ (𝑎𝑏𝑏𝑐)) → ¬ (𝑎𝑏𝑏𝑐))
395, 38pm2.21dd 621 . . . . 5 (((𝑎𝐴𝑏𝐴𝑐𝐴) ∧ (𝑎𝑏𝑏𝑐)) → 𝑎 E 𝑐)
404, 39sylan2b 287 . . . 4 (((𝑎𝐴𝑏𝐴𝑐𝐴) ∧ (𝑎 E 𝑏𝑏 E 𝑐)) → 𝑎 E 𝑐)
4140ex 115 . . 3 ((𝑎𝐴𝑏𝐴𝑐𝐴) → ((𝑎 E 𝑏𝑏 E 𝑐) → 𝑎 E 𝑐))
4241rgen3 2577 . 2 𝑎𝐴𝑏𝐴𝑐𝐴 ((𝑎 E 𝑏𝑏 E 𝑐) → 𝑎 E 𝑐)
43 df-wetr 4352 . 2 ( E We 𝐴 ↔ ( E Fr 𝐴 ∧ ∀𝑎𝐴𝑏𝐴𝑐𝐴 ((𝑎 E 𝑏𝑏 E 𝑐) → 𝑎 E 𝑐)))
441, 42, 43mpbir2an 944 1 E We 𝐴
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 709  w3a 980   = wceq 1364  wcel 2160  wral 2468  {crab 2472  c0 3437  {csn 3607  {cpr 3608   class class class wbr 4018   E cep 4305   Fr wfr 4346   We wwe 4348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-eprel 4307  df-frfor 4349  df-frind 4350  df-wetr 4352
This theorem is referenced by:  reg3exmid  4597
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