Theorem ordn2lp 6406

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 6404	. 2 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
2		ordtr 6400	. . 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 198	1 ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∈ wcel 2106  Tr wtr 5265  Ord word 6385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-fr 5641  df-we 5643  df-ord 6389

This theorem is referenced by:  ordtri1  6419  ordnbtwn  6479  suc11  6493  smoord  8404  unblem1  9326  cantnfp1lem3  9718  cardprclem  10017  nosepssdm  27746  slerec  27879