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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 7500 | . 2 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴) | |
2 | ssid 3989 | . . 3 ⊢ ∪ 𝐴 ⊆ ∪ 𝐴 | |
3 | ordunisssuc 6293 | . . 3 ⊢ ((𝐴 ⊆ On ∧ Ord ∪ 𝐴) → (∪ 𝐴 ⊆ ∪ 𝐴 ↔ 𝐴 ⊆ suc ∪ 𝐴)) | |
4 | 2, 3 | mpbii 235 | . 2 ⊢ ((𝐴 ⊆ On ∧ Ord ∪ 𝐴) → 𝐴 ⊆ suc ∪ 𝐴) |
5 | 1, 4 | mpdan 685 | 1 ⊢ (𝐴 ⊆ On → 𝐴 ⊆ suc ∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ⊆ wss 3936 ∪ cuni 4838 Ord word 6190 Oncon0 6191 suc csuc 6193 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-tr 5173 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-ord 6194 df-on 6195 df-suc 6197 |
This theorem is referenced by: ordsucuni 7544 tz9.12lem3 9218 onssnum 9466 dfac12lem2 9570 ackbij1lem16 9657 cfslb2n 9690 hsmexlem1 9848 |
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