Theorem ordn2lp 6384

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 6382	. 2 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
2		ordtr 6378	. . 3 ⊢ (Ord 𝐴 → Tr 𝐴)
3		trel 5274	. . 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 197	1 ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∈ wcel 2105  Tr wtr 5265  Ord word 6363

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-fr 5631  df-we 5633  df-ord 6367

This theorem is referenced by:  ordtri1  6397  ordnbtwn  6457  suc11  6471  smoord  8371  unblem1  9301  cantnfp1lem3  9681  cardprclem  9980  nosepssdm  27533  slerec  27666