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Mirrors > Home > MPE Home > Th. List > onssneli | 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 |
---|---|
onssneli | ⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3973 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐵)) | |
2 | on.1 | . . . . 5 ⊢ 𝐴 ∈ On | |
3 | 2 | oneli 6490 | . . . 4 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ On) |
4 | eloni 6386 | . . . 4 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
5 | ordirr 6394 | . . . 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) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2099 ⊆ wss 3947 Ord word 6375 Oncon0 6376 |
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 5304 ax-nul 5311 ax-pr 5433 |
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 3420 df-v 3464 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-tr 5271 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-ord 6379 df-on 6380 |
This theorem is referenced by: cofcutr 27941 onsucconni 36149 |
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