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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 2636 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
2 | 1 | sucid 4268 | . . . . . . . 8 ⊢ 𝑥 ∈ suc 𝑥 |
3 | elunii 3680 | . . . . . . . 8 ⊢ ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ 𝐴) → 𝑥 ∈ ∪ 𝐴) | |
4 | 2, 3 | mpan 416 | . . . . . . 7 ⊢ (suc 𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐴) |
5 | 4 | imim2i 12 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐴)) |
6 | 5 | alimi 1396 | . . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐴)) |
7 | df-ral 2375 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) | |
8 | dfss2 3028 | . . . . 5 ⊢ (𝐴 ⊆ ∪ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐴)) | |
9 | 6, 7, 8 | 3imtr4i 200 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → 𝐴 ⊆ ∪ 𝐴) |
10 | 9 | a1i 9 | . . 3 ⊢ (Ord 𝐴 → (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → 𝐴 ⊆ ∪ 𝐴)) |
11 | orduniss 4276 | . . 3 ⊢ (Ord 𝐴 → ∪ 𝐴 ⊆ 𝐴) | |
12 | 10, 11 | jctird 311 | . 2 ⊢ (Ord 𝐴 → (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → (𝐴 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝐴))) |
13 | eqss 3054 | . 2 ⊢ (𝐴 = ∪ 𝐴 ↔ (𝐴 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝐴)) | |
14 | 12, 13 | syl6ibr 161 | 1 ⊢ (Ord 𝐴 → (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → 𝐴 = ∪ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1294 = wceq 1296 ∈ wcel 1445 ∀wral 2370 ⊆ wss 3013 ∪ cuni 3675 Ord word 4213 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-v 2635 df-un 3017 df-in 3019 df-ss 3026 df-sn 3472 df-uni 3676 df-tr 3959 df-iord 4217 df-suc 4222 |
This theorem is referenced by: (None) |
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