Theorem ordn2lp 2219

Description: An ordinal class cannot an element of one of its members. Variant of first part of Theorem 2.2(vii) of [BellMachover] p. 469.

Assertion

Ref	Expression
ordn2lp	⊢ (Ord A → ¬ (A ∈ B ∧ B ∈ A))

Proof of Theorem ordn2lp

Step	Hyp	Ref	Expression
1		ordeirr 2217	. 2 ⊢ (Ord A → ¬ A ∈ A)
2		ordtr 2213	. . 3 ⊢ (Ord A → Tr A)
3		trel 2048	. . 3 ⊢ (Tr A → ((A ∈ B ∧ B ∈ A) → A ∈ A))
4	2, 3	syl 12	. 2 ⊢ (Ord A → ((A ∈ B ∧ B ∈ A) → A ∈ A))
5	1, 4	mtod 95	1 ⊢ (Ord A → ¬ (A ∈ B ∧ B ∈ A))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   ∈ wcel 1092  Tr wtr 2041  Ord word 2198

This theorem is referenced by:  ordtri1 2231  ordnbtwn 2316  suc11 2341  unblem1 3431

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-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-fr 2169  df-we 2186  df-ord 2202

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