Theorem onssnel2i 6464

Description: An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994.)

Hypothesis

Ref	Expression
on.1	⊢ 𝐴 ∈ On

Assertion

Ref	Expression
onssnel2i	⊢ (𝐵 ⊆ 𝐴 → ¬ 𝐴 ∈ 𝐵)

Proof of Theorem onssnel2i

Step	Hyp	Ref	Expression
1		on.1	. . 3 ⊢ 𝐴 ∈ On
2	1	onirri 6460	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
3		ssel 3930	. 2 ⊢ (𝐵 ⊆ 𝐴 → (𝐴 ∈ 𝐵 → 𝐴 ∈ 𝐴))
4	2, 3	mtoi 201	1 ⊢ (𝐵 ⊆ 𝐴 → ¬ 𝐴 ∈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2142   ⊆ wss 3904  Oncon0 6346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-ord 6349  df-on 6350

This theorem is referenced by: (None)