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Mirrors > Home > MPE Home > Th. List > onunisuci | Structured version Visualization version 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 6171 | . 2 ⊢ Tr 𝐴 |
3 | 1 | elexi 3456 | . . 3 ⊢ 𝐴 ∈ V |
4 | 3 | unisuc 6142 | . 2 ⊢ (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴) |
5 | 2, 4 | mpbi 231 | 1 ⊢ ∪ suc 𝐴 = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1522 ∈ wcel 2081 ∪ cuni 4745 Tr wtr 5063 Oncon0 6066 suc csuc 6068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rex 3111 df-v 3439 df-un 3864 df-in 3866 df-ss 3874 df-sn 4473 df-pr 4475 df-uni 4746 df-tr 5064 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-ord 6069 df-on 6070 df-suc 6072 |
This theorem is referenced by: rankuni 9138 onsucconni 33394 onsucsuccmpi 33400 finxp1o 34204 |
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