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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 6479 | . 2 ⊢ ¬ 𝐴 ∈ 𝐴 |
3 | ssel 3973 | . 2 ⊢ (𝐵 ⊆ 𝐴 → (𝐴 ∈ 𝐵 → 𝐴 ∈ 𝐴)) | |
4 | 2, 3 | mtoi 198 | 1 ⊢ (𝐵 ⊆ 𝐴 → ¬ 𝐴 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2099 ⊆ wss 3947 Oncon0 6366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3421 df-v 3465 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-br 5145 df-opab 5207 df-tr 5262 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-ord 6369 df-on 6370 |
This theorem is referenced by: (None) |
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