Theorem onssneli 6466

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 3968	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐵))
2		on.1	. . . . 5 ⊢ 𝐴 ∈ On
3	2	oneli 6464	. . . 4 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ On)
4		eloni 6360	. . . 4 ⊢ (𝐵 ∈ On → Ord 𝐵)
5		ordirr 6368	. . . 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 189	1 ⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2106   ⊆ wss 3941  Ord word 6349  Oncon0 6350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3430  df-v 3472  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4520  df-pw 4595  df-sn 4620  df-pr 4622  df-op 4626  df-uni 4899  df-br 5139  df-opab 5201  df-tr 5256  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-ord 6353  df-on 6354

This theorem is referenced by:  cofcutr  27326  onsucconni  35110