Theorem onssneli 6500

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 3977	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐵))
2		on.1	. . . . 5 ⊢ 𝐴 ∈ On
3	2	oneli 6498	. . . 4 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ On)
4		eloni 6394	. . . 4 ⊢ (𝐵 ∈ On → Ord 𝐵)
5		ordirr 6402	. . . 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 2108   ⊆ wss 3951  Ord word 6383  Oncon0 6384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388

This theorem is referenced by:  cofcutr  27958  onsucconni  36438