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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 4449 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On) | |
4 | 1, 2, 3 | mp2an 423 | 1 ⊢ (𝐴 ∪ 𝐵) ∈ On |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2128 ∪ cun 3100 Oncon0 4323 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pr 4169 ax-un 4393 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-sn 3566 df-pr 3567 df-uni 3773 df-tr 4063 df-iord 4326 df-on 4328 |
This theorem is referenced by: (None) |
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