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Mirrors > Home > ILE Home > Th. List > onun2 | GIF version |
Description: The union of two ordinal numbers is an ordinal number. (Contributed by Jim Kingdon, 25-Jul-2019.) |
Ref | Expression |
---|---|
onun2 | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prssi 3617 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝐴, 𝐵} ⊆ On) | |
2 | prexg 4062 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝐴, 𝐵} ∈ V) | |
3 | ssonuni 4333 | . . . 4 ⊢ ({𝐴, 𝐵} ∈ V → ({𝐴, 𝐵} ⊆ On → ∪ {𝐴, 𝐵} ∈ On)) | |
4 | 2, 3 | syl 14 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ({𝐴, 𝐵} ⊆ On → ∪ {𝐴, 𝐵} ∈ On)) |
5 | uniprg 3690 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) | |
6 | 5 | eleq1d 2163 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∪ {𝐴, 𝐵} ∈ On ↔ (𝐴 ∪ 𝐵) ∈ On)) |
7 | 4, 6 | sylibd 148 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ({𝐴, 𝐵} ⊆ On → (𝐴 ∪ 𝐵) ∈ On)) |
8 | 1, 7 | mpd 13 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1445 Vcvv 2633 ∪ cun 3011 ⊆ wss 3013 {cpr 3467 ∪ cuni 3675 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-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pr 4060 ax-un 4284 |
This theorem depends on definitions: df-bi 116 df-3an 929 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-un 3017 df-in 3019 df-ss 3026 df-sn 3472 df-pr 3473 df-uni 3676 df-tr 3959 df-iord 4217 df-on 4219 |
This theorem is referenced by: onun2i 4336 rdgon 6189 |
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