![]() |
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 4429 | . 2 ⊢ Tr 𝐴 |
3 | 1 | elexi 2751 | . . 3 ⊢ 𝐴 ∈ V |
4 | 3 | unisuc 4415 | . 2 ⊢ (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴) |
5 | 2, 4 | mpbi 145 | 1 ⊢ ∪ suc 𝐴 = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∈ wcel 2148 ∪ cuni 3811 Tr wtr 4103 Oncon0 4365 suc csuc 4367 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2741 df-un 3135 df-in 3137 df-ss 3144 df-sn 3600 df-pr 3601 df-uni 3812 df-tr 4104 df-iord 4368 df-on 4370 df-suc 4373 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |