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Theorem reg3exmidlemwe 4330
Description: Lemma for reg3exmid 4331. 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 4325 . 2 E Fr 𝐴
2 epel 4056 . . . . . 6 (𝑎 E 𝑏𝑎𝑏)
3 epel 4056 . . . . . 6 (𝑏 E 𝑐𝑏𝑐)
42, 3anbi12i 441 . . . . 5 ((𝑎 E 𝑏𝑏 E 𝑐) ↔ (𝑎𝑏𝑏𝑐))
5 simpr 107 . . . . . 6 (((𝑎𝐴𝑏𝐴𝑐𝐴) ∧ (𝑎𝑏𝑏𝑐)) → (𝑎𝑏𝑏𝑐))
6 elirr 4293 . . . . . . . 8 ¬ {∅} ∈ {∅}
7 simprr 492 . . . . . . . . . 10 (((𝑎𝐴𝑏𝐴𝑐𝐴) ∧ (𝑎𝑏𝑏𝑐)) → 𝑏𝑐)
8 noel 3255 . . . . . . . . . . . . 13 ¬ 𝑎 ∈ ∅
9 eleq2 2117 . . . . . . . . . . . . 13 (𝑏 = ∅ → (𝑎𝑏𝑎 ∈ ∅))
108, 9mtbiri 610 . . . . . . . . . . . 12 (𝑏 = ∅ → ¬ 𝑎𝑏)
11 simprl 491 . . . . . . . . . . . 12 (((𝑎𝐴𝑏𝐴𝑐𝐴) ∧ (𝑎𝑏𝑏𝑐)) → 𝑎𝑏)
1210, 11nsyl3 566 . . . . . . . . . . 11 (((𝑎𝐴𝑏𝐴𝑐𝐴) ∧ (𝑎𝑏𝑏𝑐)) → ¬ 𝑏 = ∅)
13 elrabi 2717 . . . . . . . . . . . . . . . 16 (𝑏 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))} → 𝑏 ∈ {∅, {∅}})
14 reg3exmidlemwe.a . . . . . . . . . . . . . . . 16 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
1513, 14eleq2s 2148 . . . . . . . . . . . . . . 15 (𝑏𝐴𝑏 ∈ {∅, {∅}})
16 elpri 3425 . . . . . . . . . . . . . . 15 (𝑏 ∈ {∅, {∅}} → (𝑏 = ∅ ∨ 𝑏 = {∅}))
1715, 16syl 14 . . . . . . . . . . . . . 14 (𝑏𝐴 → (𝑏 = ∅ ∨ 𝑏 = {∅}))
1817orcomd 658 . . . . . . . . . . . . 13 (𝑏𝐴 → (𝑏 = {∅} ∨ 𝑏 = ∅))
19183ad2ant2 937 . . . . . . . . . . . 12 ((𝑎𝐴𝑏𝐴𝑐𝐴) → (𝑏 = {∅} ∨ 𝑏 = ∅))
2019adantr 265 . . . . . . . . . . 11 (((𝑎𝐴𝑏𝐴𝑐𝐴) ∧ (𝑎𝑏𝑏𝑐)) → (𝑏 = {∅} ∨ 𝑏 = ∅))
2112, 20ecased 1255 . . . . . . . . . 10 (((𝑎𝐴𝑏𝐴𝑐𝐴) ∧ (𝑎𝑏𝑏𝑐)) → 𝑏 = {∅})
22 noel 3255 . . . . . . . . . . . . 13 ¬ 𝑏 ∈ ∅
23 eleq2 2117 . . . . . . . . . . . . 13 (𝑐 = ∅ → (𝑏𝑐𝑏 ∈ ∅))
2422, 23mtbiri 610 . . . . . . . . . . . 12 (𝑐 = ∅ → ¬ 𝑏𝑐)
2524, 7nsyl3 566 . . . . . . . . . . 11 (((𝑎𝐴𝑏𝐴𝑐𝐴) ∧ (𝑎𝑏𝑏𝑐)) → ¬ 𝑐 = ∅)
26 elrabi 2717 . . . . . . . . . . . . . . . 16 (𝑐 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))} → 𝑐 ∈ {∅, {∅}})
2726, 14eleq2s 2148 . . . . . . . . . . . . . . 15 (𝑐𝐴𝑐 ∈ {∅, {∅}})
28 vex 2577 . . . . . . . . . . . . . . . 16 𝑐 ∈ V
2928elpr 3423 . . . . . . . . . . . . . . 15 (𝑐 ∈ {∅, {∅}} ↔ (𝑐 = ∅ ∨ 𝑐 = {∅}))
3027, 29sylib 131 . . . . . . . . . . . . . 14 (𝑐𝐴 → (𝑐 = ∅ ∨ 𝑐 = {∅}))
3130orcomd 658 . . . . . . . . . . . . 13 (𝑐𝐴 → (𝑐 = {∅} ∨ 𝑐 = ∅))
32313ad2ant3 938 . . . . . . . . . . . 12 ((𝑎𝐴𝑏𝐴𝑐𝐴) → (𝑐 = {∅} ∨ 𝑐 = ∅))
3332adantr 265 . . . . . . . . . . 11 (((𝑎𝐴𝑏𝐴𝑐𝐴) ∧ (𝑎𝑏𝑏𝑐)) → (𝑐 = {∅} ∨ 𝑐 = ∅))
3425, 33ecased 1255 . . . . . . . . . 10 (((𝑎𝐴𝑏𝐴𝑐𝐴) ∧ (𝑎𝑏𝑏𝑐)) → 𝑐 = {∅})
357, 21, 343eltr3d 2136 . . . . . . . . 9 (((𝑎𝐴𝑏𝐴𝑐𝐴) ∧ (𝑎𝑏𝑏𝑐)) → {∅} ∈ {∅})
3635ex 112 . . . . . . . 8 ((𝑎𝐴𝑏𝐴𝑐𝐴) → ((𝑎𝑏𝑏𝑐) → {∅} ∈ {∅}))
376, 36mtoi 600 . . . . . . 7 ((𝑎𝐴𝑏𝐴𝑐𝐴) → ¬ (𝑎𝑏𝑏𝑐))
3837adantr 265 . . . . . 6 (((𝑎𝐴𝑏𝐴𝑐𝐴) ∧ (𝑎𝑏𝑏𝑐)) → ¬ (𝑎𝑏𝑏𝑐))
395, 38pm2.21dd 560 . . . . 5 (((𝑎𝐴𝑏𝐴𝑐𝐴) ∧ (𝑎𝑏𝑏𝑐)) → 𝑎 E 𝑐)
404, 39sylan2b 275 . . . 4 (((𝑎𝐴𝑏𝐴𝑐𝐴) ∧ (𝑎 E 𝑏𝑏 E 𝑐)) → 𝑎 E 𝑐)
4140ex 112 . . 3 ((𝑎𝐴𝑏𝐴𝑐𝐴) → ((𝑎 E 𝑏𝑏 E 𝑐) → 𝑎 E 𝑐))
4241rgen3 2423 . 2 𝑎𝐴𝑏𝐴𝑐𝐴 ((𝑎 E 𝑏𝑏 E 𝑐) → 𝑎 E 𝑐)
43 df-wetr 4098 . 2 ( E We 𝐴 ↔ ( E Fr 𝐴 ∧ ∀𝑎𝐴𝑏𝐴𝑐𝐴 ((𝑎 E 𝑏𝑏 E 𝑐) → 𝑎 E 𝑐)))
441, 42, 43mpbir2an 860 1 E We 𝐴
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wo 639  w3a 896   = wceq 1259  wcel 1409  wral 2323  {crab 2327  c0 3251  {csn 3402  {cpr 3403   class class class wbr 3791   E cep 4051   Fr wfr 4092   We wwe 4094
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-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-setind 4289
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rab 2332  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-eprel 4053  df-frfor 4095  df-frind 4096  df-wetr 4098
This theorem is referenced by:  reg3exmid  4331
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