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Mirrors > Home > ILE Home > Th. List > onun2i | GIF version |
Description: The union of two ordinal numbers is an ordinal number. (Contributed by NM, 13-Jun-1994.) (Constructive proof by Jim Kingdon, 25-Jul-2019.) |
Ref | Expression |
---|---|
onun2i.1 | ⊢ 𝐴 ∈ On |
onun2i.2 | ⊢ 𝐵 ∈ On |
Ref | Expression |
---|---|
onun2i | ⊢ (𝐴 ∪ 𝐵) ∈ On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onun2i.1 | . 2 ⊢ 𝐴 ∈ On | |
2 | onun2i.2 | . 2 ⊢ 𝐵 ∈ On | |
3 | onun2 4344 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On) | |
4 | 1, 2, 3 | mp2an 420 | 1 ⊢ (𝐴 ∪ 𝐵) ∈ On |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1448 ∪ cun 3019 Oncon0 4223 |
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 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pr 4069 ax-un 4293 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-un 3025 df-in 3027 df-ss 3034 df-sn 3480 df-pr 3481 df-uni 3684 df-tr 3967 df-iord 4226 df-on 4228 |
This theorem is referenced by: (None) |
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