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Mirrors > Home > ILE Home > Th. List > oneluni | GIF version |
Description: An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994.) |
Ref | Expression |
---|---|
on.1 | ⊢ 𝐴 ∈ On |
Ref | Expression |
---|---|
oneluni | ⊢ (𝐵 ∈ 𝐴 → (𝐴 ∪ 𝐵) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | on.1 | . . 3 ⊢ 𝐴 ∈ On | |
2 | 1 | onelssi 4311 | . 2 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴) |
3 | ssequn2 3215 | . 2 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∪ 𝐵) = 𝐴) | |
4 | 2, 3 | sylib 121 | 1 ⊢ (𝐵 ∈ 𝐴 → (𝐴 ∪ 𝐵) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1314 ∈ wcel 1463 ∪ cun 3035 ⊆ wss 3037 Oncon0 4245 |
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 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rex 2396 df-v 2659 df-un 3041 df-in 3043 df-ss 3050 df-uni 3703 df-tr 3987 df-iord 4248 df-on 4250 |
This theorem is referenced by: (None) |
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