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Mirrors > Home > ILE Home > Th. List > ordsson | GIF version |
Description: Any ordinal class is a subclass of the class of ordinal numbers. Corollary 7.15 of [TakeutiZaring] p. 38. (Contributed by NM, 18-May-1994.) |
Ref | Expression |
---|---|
ordsson | ⊢ (Ord 𝐴 → 𝐴 ⊆ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordelon 4313 | . . 3 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On) | |
2 | 1 | ex 114 | . 2 ⊢ (Ord 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ On)) |
3 | 2 | ssrdv 3108 | 1 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1481 ⊆ wss 3076 Ord word 4292 Oncon0 4293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-in 3082 df-ss 3089 df-uni 3745 df-tr 4035 df-iord 4296 df-on 4298 |
This theorem is referenced by: onss 4417 orduni 4419 iordsmo 6202 tfrlemi14d 6238 tfr1onlemssrecs 6244 tfri1dALT 6256 tfrcllemssrecs 6257 ordiso2 6928 |
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