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| Mirrors > Home > MPE Home > Th. List > onsucuni | Structured version Visualization version GIF version | ||
| Description: A class of ordinal numbers is a subclass of the successor of its union. Similar to Proposition 7.26 of [TakeutiZaring] p. 41. (Contributed by NM, 19-Sep-2003.) |
| Ref | Expression |
|---|---|
| onsucuni | ⊢ (𝐴 ⊆ On → 𝐴 ⊆ suc ∪ 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssorduni 7799 | . 2 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴) | |
| 2 | ssid 4006 | . . 3 ⊢ ∪ 𝐴 ⊆ ∪ 𝐴 | |
| 3 | ordunisssuc 6490 | . . 3 ⊢ ((𝐴 ⊆ On ∧ Ord ∪ 𝐴) → (∪ 𝐴 ⊆ ∪ 𝐴 ↔ 𝐴 ⊆ suc ∪ 𝐴)) | |
| 4 | 2, 3 | mpbii 233 | . 2 ⊢ ((𝐴 ⊆ On ∧ Ord ∪ 𝐴) → 𝐴 ⊆ suc ∪ 𝐴) |
| 5 | 1, 4 | mpdan 687 | 1 ⊢ (𝐴 ⊆ On → 𝐴 ⊆ suc ∪ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ⊆ wss 3951 ∪ 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 |
| 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: ordsucuni 7849 cofon1 8710 cofon2 8711 naddcllem 8714 tz9.12lem3 9829 onssnum 10080 dfac12lem2 10185 ackbij1lem16 10274 cfslb2n 10308 hsmexlem1 10466 noeta2 27829 etasslt2 27859 zs12bday 28424 cantnfub2 43335 onsucunifi 43383 |
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