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Mirrors > Home > ILE Home > Th. List > onelss | GIF version |
Description: An element of an ordinal number is a subset of the number. (Contributed by NM, 5-Jun-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
onelss | ⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 4226 | . 2 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
2 | ordelss 4230 | . . 3 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴) | |
3 | 2 | ex 114 | . 2 ⊢ (Ord 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) |
4 | 1, 3 | syl 14 | 1 ⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1445 ⊆ wss 3013 Ord word 4213 Oncon0 4214 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-in 3019 df-ss 3026 df-uni 3676 df-tr 3959 df-iord 4217 df-on 4219 |
This theorem is referenced by: onelssi 4280 ssorduni 4332 onsucelsucr 4353 tfisi 4430 tfrlem9 6122 nntri2or2 6299 phpelm 6662 |
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