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Mirrors > Home > ILE Home > Th. List > ordunisuc2r | GIF version |
Description: An ordinal which contains the successor of each of its members is equal to its union. (Contributed by Jim Kingdon, 14-Nov-2018.) |
Ref | Expression |
---|---|
ordunisuc2r | ⊢ (Ord 𝐴 → (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → 𝐴 = ∪ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2605 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
2 | 1 | sucid 4174 | . . . . . . . 8 ⊢ 𝑥 ∈ suc 𝑥 |
3 | elunii 3608 | . . . . . . . 8 ⊢ ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ 𝐴) → 𝑥 ∈ ∪ 𝐴) | |
4 | 2, 3 | mpan 415 | . . . . . . 7 ⊢ (suc 𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐴) |
5 | 4 | imim2i 12 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐴)) |
6 | 5 | alimi 1385 | . . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐴)) |
7 | df-ral 2354 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) | |
8 | dfss2 2989 | . . . . 5 ⊢ (𝐴 ⊆ ∪ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐴)) | |
9 | 6, 7, 8 | 3imtr4i 199 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → 𝐴 ⊆ ∪ 𝐴) |
10 | 9 | a1i 9 | . . 3 ⊢ (Ord 𝐴 → (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → 𝐴 ⊆ ∪ 𝐴)) |
11 | orduniss 4182 | . . 3 ⊢ (Ord 𝐴 → ∪ 𝐴 ⊆ 𝐴) | |
12 | 10, 11 | jctird 310 | . 2 ⊢ (Ord 𝐴 → (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → (𝐴 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝐴))) |
13 | eqss 3015 | . 2 ⊢ (𝐴 = ∪ 𝐴 ↔ (𝐴 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝐴)) | |
14 | 12, 13 | syl6ibr 160 | 1 ⊢ (Ord 𝐴 → (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → 𝐴 = ∪ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∀wal 1283 = wceq 1285 ∈ wcel 1434 ∀wral 2349 ⊆ wss 2974 ∪ cuni 3603 Ord word 4119 suc csuc 4122 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1687 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ral 2354 df-v 2604 df-un 2978 df-in 2980 df-ss 2987 df-sn 3406 df-uni 3604 df-tr 3878 df-iord 4123 df-suc 4128 |
This theorem is referenced by: (None) |
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