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| Mirrors > Home > MPE Home > Th. List > onssnel2i | Structured version Visualization version GIF version | ||
| Description: An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994.) | 
| Ref | Expression | 
|---|---|
| on.1 | ⊢ 𝐴 ∈ On | 
| Ref | Expression | 
|---|---|
| onssnel2i | ⊢ (𝐵 ⊆ 𝐴 → ¬ 𝐴 ∈ 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | on.1 | . . 3 ⊢ 𝐴 ∈ On | |
| 2 | 1 | onirri 6496 | . 2 ⊢ ¬ 𝐴 ∈ 𝐴 | 
| 3 | ssel 3976 | . 2 ⊢ (𝐵 ⊆ 𝐴 → (𝐴 ∈ 𝐵 → 𝐴 ∈ 𝐴)) | |
| 4 | 2, 3 | mtoi 199 | 1 ⊢ (𝐵 ⊆ 𝐴 → ¬ 𝐴 ∈ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2107 ⊆ wss 3950 Oncon0 6383 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-tr 5259 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-ord 6386 df-on 6387 | 
| This theorem is referenced by: (None) | 
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