Theorem onssnel2i 6296

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 6292	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
3		ssel 3961	. 2 ⊢ (𝐵 ⊆ 𝐴 → (𝐴 ∈ 𝐵 → 𝐴 ∈ 𝐴))
4	2, 3	mtoi 201	1 ⊢ (𝐵 ⊆ 𝐴 → ¬ 𝐴 ∈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2110   ⊆ wss 3936  Oncon0 6186

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  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 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-tr 5166  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-ord 6189  df-on 6190

This theorem is referenced by: (None)