Theorem en3lpVD 42465

Description: Virtual deduction proof of en3lp 9372. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
en3lpVD	⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)

Proof of Theorem en3lpVD

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm2.1 894	. . 3 ⊢ (¬ {𝐴, 𝐵, 𝐶} = ∅ ∨ {𝐴, 𝐵, 𝐶} = ∅)
2		df-ne 2944	. . . . 5 ⊢ ({𝐴, 𝐵, 𝐶} ≠ ∅ ↔ ¬ {𝐴, 𝐵, 𝐶} = ∅)
3	2	bicomi 223	. . . 4 ⊢ (¬ {𝐴, 𝐵, 𝐶} = ∅ ↔ {𝐴, 𝐵, 𝐶} ≠ ∅)
4	3	orbi1i 911	. . 3 ⊢ ((¬ {𝐴, 𝐵, 𝐶} = ∅ ∨ {𝐴, 𝐵, 𝐶} = ∅) ↔ ({𝐴, 𝐵, 𝐶} ≠ ∅ ∨ {𝐴, 𝐵, 𝐶} = ∅))
5	1, 4	mpbi 229	. 2 ⊢ ({𝐴, 𝐵, 𝐶} ≠ ∅ ∨ {𝐴, 𝐵, 𝐶} = ∅)
6		zfregs2 9491	. . . 4 ⊢ ({𝐴, 𝐵, 𝐶} ≠ ∅ → ¬ ∀𝑥 ∈ {𝐴, 𝐵, 𝐶}∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥))
7		en3lplem2VD 42464	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥)))
8	7	alrimiv 1930	. . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → ∀𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥)))
9		df-ral 3069	. . . . . 6 ⊢ (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥) ↔ ∀𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥)))
10	8, 9	sylibr 233	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → ∀𝑥 ∈ {𝐴, 𝐵, 𝐶}∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥))
11	10	con3i 154	. . . 4 ⊢ (¬ ∀𝑥 ∈ {𝐴, 𝐵, 𝐶}∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦 ∈ 𝑥) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴))
12	6, 11	syl 17	. . 3 ⊢ ({𝐴, 𝐵, 𝐶} ≠ ∅ → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴))
13		idn1 42194	. . . . . . 7 ⊢ (   {𝐴, 𝐵, 𝐶} = ∅   ▶   {𝐴, 𝐵, 𝐶} = ∅   )
14		noel 4264	. . . . . . 7 ⊢ ¬ 𝐶 ∈ ∅
15		eleq2 2827	. . . . . . . . 9 ⊢ ({𝐴, 𝐵, 𝐶} = ∅ → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐶 ∈ ∅))
16	15	notbid 318	. . . . . . . 8 ⊢ ({𝐴, 𝐵, 𝐶} = ∅ → (¬ 𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ ¬ 𝐶 ∈ ∅))
17	16	biimprd 247	. . . . . . 7 ⊢ ({𝐴, 𝐵, 𝐶} = ∅ → (¬ 𝐶 ∈ ∅ → ¬ 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
18	13, 14, 17	e10 42314	. . . . . 6 ⊢ (   {𝐴, 𝐵, 𝐶} = ∅   ▶    ¬ 𝐶 ∈ {𝐴, 𝐵, 𝐶}   )
19		tpid3g 4708	. . . . . . 7 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
20	19	con3i 154	. . . . . 6 ⊢ (¬ 𝐶 ∈ {𝐴, 𝐵, 𝐶} → ¬ 𝐶 ∈ 𝐴)
21	18, 20	e1a 42247	. . . . 5 ⊢ (   {𝐴, 𝐵, 𝐶} = ∅   ▶    ¬ 𝐶 ∈ 𝐴   )
22		simp3 1137	. . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐴)
23	22	con3i 154	. . . . 5 ⊢ (¬ 𝐶 ∈ 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴))
24	21, 23	e1a 42247	. . . 4 ⊢ (   {𝐴, 𝐵, 𝐶} = ∅   ▶    ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)   )
25	24	in1 42191	. . 3 ⊢ ({𝐴, 𝐵, 𝐶} = ∅ → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴))
26	12, 25	jaoi 854	. 2 ⊢ (({𝐴, 𝐵, 𝐶} ≠ ∅ ∨ {𝐴, 𝐵, 𝐶} = ∅) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴))
27	5, 26	ax-mp 5	1 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 844   ∧ w3a 1086  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∅c0 4256  {ctp 4565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588  ax-reg 9351  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-vd1 42190  df-vd2 42198  df-vd3 42210

This theorem is referenced by: (None)