Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3584 | . . . 4 ⊢ (𝑥 ∈ ∪ On ↔ ∃𝑦 ∈ On 𝑥 ∈ 𝑦) | |
2 | onelon 4121 | . . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ On) | |
3 | 2 | rexlimiva 2428 | . . . 4 ⊢ (∃𝑦 ∈ On 𝑥 ∈ 𝑦 → 𝑥 ∈ On) |
4 | 1, 3 | sylbi 114 | . . 3 ⊢ (𝑥 ∈ ∪ On → 𝑥 ∈ On) |
5 | vex 2560 | . . . . 5 ⊢ 𝑥 ∈ V | |
6 | 5 | sucid 4154 | . . . 4 ⊢ 𝑥 ∈ suc 𝑥 |
7 | suceloni 4227 | . . . 4 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On) | |
8 | elunii 3585 | . . . 4 ⊢ ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ On) → 𝑥 ∈ ∪ On) | |
9 | 6, 7, 8 | sylancr 393 | . . 3 ⊢ (𝑥 ∈ On → 𝑥 ∈ ∪ On) |
10 | 4, 9 | impbii 117 | . 2 ⊢ (𝑥 ∈ ∪ On ↔ 𝑥 ∈ On) |
11 | 10 | eqriv 2037 | 1 ⊢ ∪ On = On |
Colors of variables: wff set class |
Syntax hints: = wceq 1243 ∈ wcel 1393 ∃wrex 2307 ∪ cuni 3580 Oncon0 4100 suc csuc 4102 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-uni 3581 df-tr 3855 df-iord 4103 df-on 4105 df-suc 4108 |
This theorem is referenced by: limon 4239 onintonm 4243 |
Copyright terms: Public domain | W3C validator |