Theorem ordn2lp 6331

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 6329	. 2 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
2		ordtr 6325	. . 3 ⊢ (Ord 𝐴 → Tr 𝐴)
3		trel 5208	. . 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 2113  Tr wtr 5200  Ord word 6310

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-tr 5201  df-eprel 5519  df-fr 5572  df-we 5574  df-ord 6314

This theorem is referenced by:  ordtri1  6344  ordnbtwn  6406  suc11  6420  smoord  8291  unblem1  9183  cantnfp1lem3  9577  cardprclem  9879  nosepssdm  27626  slerec  27761