Theorem onssneli 6502

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

Hypothesis

Ref	Expression
on.1	⊢ 𝐴 ∈ On

Assertion

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

Proof of Theorem onssneli

Step	Hyp	Ref	Expression
1		ssel 3989	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐵))
2		on.1	. . . . 5 ⊢ 𝐴 ∈ On
3	2	oneli 6500	. . . 4 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ On)
4		eloni 6396	. . . 4 ⊢ (𝐵 ∈ On → Ord 𝐵)
5		ordirr 6404	. . . 4 ⊢ (Ord 𝐵 → ¬ 𝐵 ∈ 𝐵)
6	3, 4, 5	3syl 18	. . 3 ⊢ (𝐵 ∈ 𝐴 → ¬ 𝐵 ∈ 𝐵)
7	1, 6	nsyli 157	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∈ 𝐴 → ¬ 𝐵 ∈ 𝐴))
8	7	pm2.01d 190	1 ⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2106   ⊆ wss 3963  Ord word 6385  Oncon0 6386

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-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390

This theorem is referenced by:  cofcutr  27973  onsucconni  36420