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| Mirrors > Home > MPE Home > Th. List > orduniss2 | Structured version Visualization version GIF version | ||
| Description: The union of the ordinal subsets of an ordinal number is that number. (Contributed by NM, 30-Jan-2005.) |
| Ref | Expression |
|---|---|
| orduniss2 | ⊢ (Ord 𝐴 → ∪ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-rab 3437 | . . . . 5 ⊢ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴)} | |
| 2 | incom 4209 | . . . . . 6 ⊢ ({𝑥 ∣ 𝑥 ∈ On} ∩ {𝑥 ∣ 𝑥 ⊆ 𝐴}) = ({𝑥 ∣ 𝑥 ⊆ 𝐴} ∩ {𝑥 ∣ 𝑥 ∈ On}) | |
| 3 | inab 4309 | . . . . . 6 ⊢ ({𝑥 ∣ 𝑥 ∈ On} ∩ {𝑥 ∣ 𝑥 ⊆ 𝐴}) = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴)} | |
| 4 | df-pw 4602 | . . . . . . . 8 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴} | |
| 5 | 4 | eqcomi 2746 | . . . . . . 7 ⊢ {𝑥 ∣ 𝑥 ⊆ 𝐴} = 𝒫 𝐴 |
| 6 | abid2 2879 | . . . . . . 7 ⊢ {𝑥 ∣ 𝑥 ∈ On} = On | |
| 7 | 5, 6 | ineq12i 4218 | . . . . . 6 ⊢ ({𝑥 ∣ 𝑥 ⊆ 𝐴} ∩ {𝑥 ∣ 𝑥 ∈ On}) = (𝒫 𝐴 ∩ On) |
| 8 | 2, 3, 7 | 3eqtr3i 2773 | . . . . 5 ⊢ {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴)} = (𝒫 𝐴 ∩ On) |
| 9 | 1, 8 | eqtri 2765 | . . . 4 ⊢ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = (𝒫 𝐴 ∩ On) |
| 10 | ordpwsuc 7835 | . . . 4 ⊢ (Ord 𝐴 → (𝒫 𝐴 ∩ On) = suc 𝐴) | |
| 11 | 9, 10 | eqtrid 2789 | . . 3 ⊢ (Ord 𝐴 → {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = suc 𝐴) |
| 12 | 11 | unieqd 4920 | . 2 ⊢ (Ord 𝐴 → ∪ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = ∪ suc 𝐴) |
| 13 | ordunisuc 7852 | . 2 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴) | |
| 14 | 12, 13 | eqtrd 2777 | 1 ⊢ (Ord 𝐴 → ∪ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2714 {crab 3436 ∩ cin 3950 ⊆ wss 3951 𝒫 cpw 4600 ∪ cuni 4907 Ord word 6383 Oncon0 6384 suc csuc 6386 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 df-suc 6390 |
| This theorem is referenced by: (None) |
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