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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 4278 | . 2 ⊢ Tr 𝐴 |
3 | 1 | elexi 2645 | . . 3 ⊢ 𝐴 ∈ V |
4 | 3 | unisuc 4264 | . 2 ⊢ (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴) |
5 | 2, 4 | mpbi 144 | 1 ⊢ ∪ suc 𝐴 = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1296 ∈ wcel 1445 ∪ cuni 3675 Tr wtr 3958 Oncon0 4214 suc csuc 4216 |
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 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-un 3017 df-in 3019 df-ss 3026 df-sn 3472 df-pr 3473 df-uni 3676 df-tr 3959 df-iord 4217 df-on 4219 df-suc 4222 |
This theorem is referenced by: (None) |
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