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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 3073 | . . . . 5 ⊢ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴)} | |
2 | incom 4132 | . . . . . 6 ⊢ ({𝑥 ∣ 𝑥 ∈ On} ∩ {𝑥 ∣ 𝑥 ⊆ 𝐴}) = ({𝑥 ∣ 𝑥 ⊆ 𝐴} ∩ {𝑥 ∣ 𝑥 ∈ On}) | |
3 | inab 4231 | . . . . . 6 ⊢ ({𝑥 ∣ 𝑥 ∈ On} ∩ {𝑥 ∣ 𝑥 ⊆ 𝐴}) = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴)} | |
4 | df-pw 4532 | . . . . . . . 8 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴} | |
5 | 4 | eqcomi 2748 | . . . . . . 7 ⊢ {𝑥 ∣ 𝑥 ⊆ 𝐴} = 𝒫 𝐴 |
6 | abid2 2882 | . . . . . . 7 ⊢ {𝑥 ∣ 𝑥 ∈ On} = On | |
7 | 5, 6 | ineq12i 4142 | . . . . . 6 ⊢ ({𝑥 ∣ 𝑥 ⊆ 𝐴} ∩ {𝑥 ∣ 𝑥 ∈ On}) = (𝒫 𝐴 ∩ On) |
8 | 2, 3, 7 | 3eqtr3i 2775 | . . . . 5 ⊢ {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴)} = (𝒫 𝐴 ∩ On) |
9 | 1, 8 | eqtri 2767 | . . . 4 ⊢ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = (𝒫 𝐴 ∩ On) |
10 | ordpwsuc 7616 | . . . 4 ⊢ (Ord 𝐴 → (𝒫 𝐴 ∩ On) = suc 𝐴) | |
11 | 9, 10 | eqtrid 2791 | . . 3 ⊢ (Ord 𝐴 → {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = suc 𝐴) |
12 | 11 | unieqd 4850 | . 2 ⊢ (Ord 𝐴 → ∪ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = ∪ suc 𝐴) |
13 | ordunisuc 7633 | . 2 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴) | |
14 | 12, 13 | eqtrd 2779 | 1 ⊢ (Ord 𝐴 → ∪ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 {cab 2716 {crab 3068 ∩ cin 3882 ⊆ wss 3883 𝒫 cpw 4530 ∪ cuni 4836 Ord word 6233 Oncon0 6234 suc csuc 6236 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-11 2160 ax-ext 2710 ax-sep 5209 ax-nul 5216 ax-pr 5339 ax-un 7545 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2717 df-cleq 2731 df-clel 2818 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3425 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4255 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-tr 5179 df-eprel 5478 df-po 5486 df-so 5487 df-fr 5527 df-we 5529 df-ord 6237 df-on 6238 df-suc 6240 |
This theorem is referenced by: (None) |
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