Theorem efrirr 2180

Description: Irreflexivitiy of the epsilon relation: a class founded by epsilon is not a member of itself.

Assertion

Ref	Expression
efrirr	⊢ (E Fr A → ¬ A ∈ A)

Proof of Theorem efrirr

Step	Hyp	Ref	Expression
1		freq2 2175	. . . . 5 ⊢ (x = A → (E Fr x ↔ E Fr A))
2		eleq1 1149	. . . . . . 7 ⊢ (x = A → (x ∈ x ↔ A ∈ x))
3		eleq2 1150	. . . . . . 7 ⊢ (x = A → (A ∈ x ↔ A ∈ A))
4	2, 3	bitrd 406	. . . . . 6 ⊢ (x = A → (x ∈ x ↔ A ∈ A))
5	4	negbid 463	. . . . 5 ⊢ (x = A → (¬ x ∈ x ↔ ¬ A ∈ A))
6	1, 5	imbi12d 474	. . . 4 ⊢ (x = A → ((E Fr x → ¬ x ∈ x) ↔ (E Fr A → ¬ A ∈ A)))
7		frirr 2176	. . . . . . 7 ⊢ ((E Fr x ∧ x ∈ x) → ¬ xEx)
8	7	exp 291	. . . . . 6 ⊢ (E Fr x → (x ∈ x → ¬ xEx))
9		epel 2124	. . . . . . 7 ⊢ (xEx ↔ x ∈ x)
10	9	negbii 162	. . . . . 6 ⊢ (¬ xEx ↔ ¬ x ∈ x)
11	8, 10	syl6ib 185	. . . . 5 ⊢ (E Fr x → (x ∈ x → ¬ x ∈ x))
12	11	pm2.01d 81	. . . 4 ⊢ (E Fr x → ¬ x ∈ x)
13	6, 12	vtoclg 1383	. . 3 ⊢ (A ∈ A → (E Fr A → ¬ A ∈ A))
14	13	com12 13	. 2 ⊢ (E Fr A → (A ∈ A → ¬ A ∈ A))
15	14	pm2.01d 81	1 ⊢ (E Fr A → ¬ A ∈ A)

Syntax hints:  ¬ wn 1   → wi 2   ∈ wel 803   = wceq 1091   ∈ wcel 1092   class class class wbr 2054  Ecep 2056   Fr wfr 2061

This theorem is referenced by:  tz7.2 2183  ordeirr 2217

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-eprel 2122  df-fr 2169

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