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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 5750 | . 2 ⊢ Tr 𝐴 |
3 | 1 | elexi 3186 | . . 3 ⊢ 𝐴 ∈ V |
4 | 3 | unisuc 5718 | . 2 ⊢ (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴) |
5 | 2, 4 | mpbi 219 | 1 ⊢ ∪ suc 𝐴 = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 ∪ cuni 4372 Tr wtr 4680 Oncon0 5640 suc csuc 5642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-un 3545 df-in 3547 df-ss 3554 df-sn 4126 df-pr 4128 df-uni 4373 df-tr 4681 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 df-suc 5646 |
This theorem is referenced by: rankuni 8609 onsucconi 31606 onsucsuccmpi 31612 finxp1o 32405 |
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