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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 2752 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
2 | 1 | sucid 4429 | . . . . . . . 8 ⊢ 𝑥 ∈ suc 𝑥 |
3 | elunii 3826 | . . . . . . . 8 ⊢ ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ 𝐴) → 𝑥 ∈ ∪ 𝐴) | |
4 | 2, 3 | mpan 424 | . . . . . . 7 ⊢ (suc 𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐴) |
5 | 4 | imim2i 12 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐴)) |
6 | 5 | alimi 1465 | . . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐴)) |
7 | df-ral 2470 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) | |
8 | dfss2 3156 | . . . . 5 ⊢ (𝐴 ⊆ ∪ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐴)) | |
9 | 6, 7, 8 | 3imtr4i 201 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → 𝐴 ⊆ ∪ 𝐴) |
10 | 9 | a1i 9 | . . 3 ⊢ (Ord 𝐴 → (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → 𝐴 ⊆ ∪ 𝐴)) |
11 | orduniss 4437 | . . 3 ⊢ (Ord 𝐴 → ∪ 𝐴 ⊆ 𝐴) | |
12 | 10, 11 | jctird 317 | . 2 ⊢ (Ord 𝐴 → (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → (𝐴 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝐴))) |
13 | eqss 3182 | . 2 ⊢ (𝐴 = ∪ 𝐴 ↔ (𝐴 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝐴)) | |
14 | 12, 13 | imbitrrdi 162 | 1 ⊢ (Ord 𝐴 → (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → 𝐴 = ∪ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∀wal 1361 = wceq 1363 ∈ wcel 2158 ∀wral 2465 ⊆ wss 3141 ∪ cuni 3821 Ord word 4374 suc csuc 4377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-sn 3610 df-uni 3822 df-tr 4114 df-iord 4378 df-suc 4383 |
This theorem is referenced by: (None) |
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