Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > onunisuci | GIF version |
Description: An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.) |
Ref | Expression |
---|---|
on.1 | ⊢ 𝐴 ∈ On |
Ref | Expression |
---|---|
onunisuci | ⊢ ∪ suc 𝐴 = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | on.1 | . . 3 ⊢ 𝐴 ∈ On | |
2 | 1 | ontrci 4410 | . 2 ⊢ Tr 𝐴 |
3 | 1 | elexi 2742 | . . 3 ⊢ 𝐴 ∈ V |
4 | 3 | unisuc 4396 | . 2 ⊢ (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴) |
5 | 2, 4 | mpbi 144 | 1 ⊢ ∪ suc 𝐴 = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 ∈ wcel 2141 ∪ cuni 3794 Tr wtr 4085 Oncon0 4346 suc csuc 4348 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-sn 3587 df-pr 3588 df-uni 3795 df-tr 4086 df-iord 4349 df-on 4351 df-suc 4354 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |