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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 7770 | . 2 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴) | |
2 | ssid 4004 | . . 3 ⊢ ∪ 𝐴 ⊆ ∪ 𝐴 | |
3 | ordunisssuc 6470 | . . 3 ⊢ ((𝐴 ⊆ On ∧ Ord ∪ 𝐴) → (∪ 𝐴 ⊆ ∪ 𝐴 ↔ 𝐴 ⊆ suc ∪ 𝐴)) | |
4 | 2, 3 | mpbii 232 | . 2 ⊢ ((𝐴 ⊆ On ∧ Ord ∪ 𝐴) → 𝐴 ⊆ suc ∪ 𝐴) |
5 | 1, 4 | mpdan 684 | 1 ⊢ (𝐴 ⊆ On → 𝐴 ⊆ suc ∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ⊆ wss 3948 ∪ cuni 4908 Ord word 6363 Oncon0 6364 suc csuc 6366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-ord 6367 df-on 6368 df-suc 6370 |
This theorem is referenced by: ordsucuni 7821 cofon1 8677 cofon2 8678 naddcllem 8681 tz9.12lem3 9790 onssnum 10041 dfac12lem2 10145 ackbij1lem16 10236 cfslb2n 10269 hsmexlem1 10427 noeta2 27629 etasslt2 27659 cantnfub2 42534 onsucunifi 42582 |
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