![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 3765 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝐴, 𝐵} ⊆ On) | |
2 | prexg 4229 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝐴, 𝐵} ∈ V) | |
3 | ssonuni 4505 | . . . 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 3839 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) | |
6 | 5 | eleq1d 2258 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∪ {𝐴, 𝐵} ∈ On ↔ (𝐴 ∪ 𝐵) ∈ On)) |
7 | 4, 6 | sylibd 149 | . 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 104 ∈ wcel 2160 Vcvv 2752 ∪ cun 3142 ⊆ wss 3144 {cpr 3608 ∪ cuni 3824 Oncon0 4381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-sn 3613 df-pr 3614 df-uni 3825 df-tr 4117 df-iord 4384 df-on 4386 |
This theorem is referenced by: onun2i 4508 rdgon 6410 |
Copyright terms: Public domain | W3C validator |