Theorem ordn2lp 6211

Description: An ordinal class cannot be an element of one of its members. Variant of first part of Theorem 2.2(vii) of [BellMachover] p. 469. (Contributed by NM, 3-Apr-1994.)

Assertion

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

Proof of Theorem ordn2lp

Step	Hyp	Ref	Expression
1		ordirr 6209	. 2 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
2		ordtr 6205	. . 3 ⊢ (Ord 𝐴 → Tr 𝐴)
3		trel 5179	. . 3 ⊢ (Tr 𝐴 → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) → 𝐴 ∈ 𝐴))
4	2, 3	syl 17	. 2 ⊢ (Ord 𝐴 → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) → 𝐴 ∈ 𝐴))
5	1, 4	mtod 200	1 ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∈ wcel 2114  Tr wtr 5172  Ord word 6190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-tr 5173  df-eprel 5465  df-fr 5514  df-we 5516  df-ord 6194

This theorem is referenced by:  ordtri1  6224  ordnbtwn  6281  suc11  6294  smoord  8002  unblem1  8770  cantnfp1lem3  9143  cardprclem  9408  nosepssdm  33190  slerec  33277