Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  en3lpVD Structured version   Visualization version   GIF version

Theorem en3lpVD 41544
Description: Virtual deduction proof of en3lp 9065. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
en3lpVD ¬ (𝐴𝐵𝐵𝐶𝐶𝐴)

Proof of Theorem en3lpVD
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pm2.1 894 . . 3 (¬ {𝐴, 𝐵, 𝐶} = ∅ ∨ {𝐴, 𝐵, 𝐶} = ∅)
2 df-ne 2991 . . . . 5 ({𝐴, 𝐵, 𝐶} ≠ ∅ ↔ ¬ {𝐴, 𝐵, 𝐶} = ∅)
32bicomi 227 . . . 4 (¬ {𝐴, 𝐵, 𝐶} = ∅ ↔ {𝐴, 𝐵, 𝐶} ≠ ∅)
43orbi1i 911 . . 3 ((¬ {𝐴, 𝐵, 𝐶} = ∅ ∨ {𝐴, 𝐵, 𝐶} = ∅) ↔ ({𝐴, 𝐵, 𝐶} ≠ ∅ ∨ {𝐴, 𝐵, 𝐶} = ∅))
51, 4mpbi 233 . 2 ({𝐴, 𝐵, 𝐶} ≠ ∅ ∨ {𝐴, 𝐵, 𝐶} = ∅)
6 zfregs2 9163 . . . 4 ({𝐴, 𝐵, 𝐶} ≠ ∅ → ¬ ∀𝑥 ∈ {𝐴, 𝐵, 𝐶}∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))
7 en3lplem2VD 41543 . . . . . . 7 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)))
87alrimiv 1928 . . . . . 6 ((𝐴𝐵𝐵𝐶𝐶𝐴) → ∀𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)))
9 df-ral 3114 . . . . . 6 (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥) ↔ ∀𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)))
108, 9sylibr 237 . . . . 5 ((𝐴𝐵𝐵𝐶𝐶𝐴) → ∀𝑥 ∈ {𝐴, 𝐵, 𝐶}∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥))
1110con3i 157 . . . 4 (¬ ∀𝑥 ∈ {𝐴, 𝐵, 𝐶}∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥) → ¬ (𝐴𝐵𝐵𝐶𝐶𝐴))
126, 11syl 17 . . 3 ({𝐴, 𝐵, 𝐶} ≠ ∅ → ¬ (𝐴𝐵𝐵𝐶𝐶𝐴))
13 idn1 41273 . . . . . . 7 (   {𝐴, 𝐵, 𝐶} = ∅   ▶   {𝐴, 𝐵, 𝐶} = ∅   )
14 noel 4250 . . . . . . 7 ¬ 𝐶 ∈ ∅
15 eleq2 2881 . . . . . . . . 9 ({𝐴, 𝐵, 𝐶} = ∅ → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐶 ∈ ∅))
1615notbid 321 . . . . . . . 8 ({𝐴, 𝐵, 𝐶} = ∅ → (¬ 𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ ¬ 𝐶 ∈ ∅))
1716biimprd 251 . . . . . . 7 ({𝐴, 𝐵, 𝐶} = ∅ → (¬ 𝐶 ∈ ∅ → ¬ 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
1813, 14, 17e10 41393 . . . . . 6 (   {𝐴, 𝐵, 𝐶} = ∅   ▶    ¬ 𝐶 ∈ {𝐴, 𝐵, 𝐶}   )
19 tpid3g 4671 . . . . . . 7 (𝐶𝐴𝐶 ∈ {𝐴, 𝐵, 𝐶})
2019con3i 157 . . . . . 6 𝐶 ∈ {𝐴, 𝐵, 𝐶} → ¬ 𝐶𝐴)
2118, 20e1a 41326 . . . . 5 (   {𝐴, 𝐵, 𝐶} = ∅   ▶    ¬ 𝐶𝐴   )
22 simp3 1135 . . . . . 6 ((𝐴𝐵𝐵𝐶𝐶𝐴) → 𝐶𝐴)
2322con3i 157 . . . . 5 𝐶𝐴 → ¬ (𝐴𝐵𝐵𝐶𝐶𝐴))
2421, 23e1a 41326 . . . 4 (   {𝐴, 𝐵, 𝐶} = ∅   ▶    ¬ (𝐴𝐵𝐵𝐶𝐶𝐴)   )
2524in1 41270 . . 3 ({𝐴, 𝐵, 𝐶} = ∅ → ¬ (𝐴𝐵𝐵𝐶𝐶𝐴))
2612, 25jaoi 854 . 2 (({𝐴, 𝐵, 𝐶} ≠ ∅ ∨ {𝐴, 𝐵, 𝐶} = ∅) → ¬ (𝐴𝐵𝐵𝐶𝐶𝐴))
275, 26ax-mp 5 1 ¬ (𝐴𝐵𝐵𝐶𝐶𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wo 844  w3a 1084  wal 1536   = wceq 1538  wex 1781  wcel 2112  wne 2990  wral 3109  c0 4246  {ctp 4532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-reg 9044  ax-inf2 9092
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-om 7565  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-vd1 41269  df-vd2 41277  df-vd3 41289
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator