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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 3958 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐵)) | |
2 | on.1 | . . . . 5 ⊢ 𝐴 ∈ On | |
3 | 2 | oneli 6291 | . . . 4 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ On) |
4 | eloni 6194 | . . . 4 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
5 | ordirr 6202 | . . . 4 ⊢ (Ord 𝐵 → ¬ 𝐵 ∈ 𝐵) | |
6 | 3, 4, 5 | 3syl 18 | . . 3 ⊢ (𝐵 ∈ 𝐴 → ¬ 𝐵 ∈ 𝐵) |
7 | 1, 6 | nsyli 160 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∈ 𝐴 → ¬ 𝐵 ∈ 𝐴)) |
8 | 7 | pm2.01d 191 | 1 ⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2105 ⊆ wss 3933 Ord word 6183 Oncon0 6184 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-ord 6187 df-on 6188 |
This theorem is referenced by: onsucconni 33682 |
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