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Mirrors > Home > ILE Home > Th. List > unon | GIF version |
Description: The class of all ordinal numbers is its own union. Exercise 11 of [TakeutiZaring] p. 40. (Contributed by NM, 12-Nov-2003.) |
Ref | Expression |
---|---|
unon | ⊢ ∪ On = On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluni2 3740 | . . . 4 ⊢ (𝑥 ∈ ∪ On ↔ ∃𝑦 ∈ On 𝑥 ∈ 𝑦) | |
2 | onelon 4306 | . . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ On) | |
3 | 2 | rexlimiva 2544 | . . . 4 ⊢ (∃𝑦 ∈ On 𝑥 ∈ 𝑦 → 𝑥 ∈ On) |
4 | 1, 3 | sylbi 120 | . . 3 ⊢ (𝑥 ∈ ∪ On → 𝑥 ∈ On) |
5 | vex 2689 | . . . . 5 ⊢ 𝑥 ∈ V | |
6 | 5 | sucid 4339 | . . . 4 ⊢ 𝑥 ∈ suc 𝑥 |
7 | suceloni 4417 | . . . 4 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On) | |
8 | elunii 3741 | . . . 4 ⊢ ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ On) → 𝑥 ∈ ∪ On) | |
9 | 6, 7, 8 | sylancr 410 | . . 3 ⊢ (𝑥 ∈ On → 𝑥 ∈ ∪ On) |
10 | 4, 9 | impbii 125 | . 2 ⊢ (𝑥 ∈ ∪ On ↔ 𝑥 ∈ On) |
11 | 10 | eqriv 2136 | 1 ⊢ ∪ On = On |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 ∃wrex 2417 ∪ cuni 3736 Oncon0 4285 suc csuc 4287 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-uni 3737 df-tr 4027 df-iord 4288 df-on 4290 df-suc 4293 |
This theorem is referenced by: limon 4429 onintonm 4433 tfri1dALT 6248 rdgon 6283 |
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