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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 7489 | . 2 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴) | |
2 | ssid 3986 | . . 3 ⊢ ∪ 𝐴 ⊆ ∪ 𝐴 | |
3 | ordunisssuc 6286 | . . 3 ⊢ ((𝐴 ⊆ On ∧ Ord ∪ 𝐴) → (∪ 𝐴 ⊆ ∪ 𝐴 ↔ 𝐴 ⊆ suc ∪ 𝐴)) | |
4 | 2, 3 | mpbii 234 | . 2 ⊢ ((𝐴 ⊆ On ∧ Ord ∪ 𝐴) → 𝐴 ⊆ suc ∪ 𝐴) |
5 | 1, 4 | mpdan 683 | 1 ⊢ (𝐴 ⊆ On → 𝐴 ⊆ suc ∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ⊆ wss 3933 ∪ cuni 4830 Ord word 6183 Oncon0 6184 suc csuc 6186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-ord 6187 df-on 6188 df-suc 6190 |
This theorem is referenced by: ordsucuni 7533 tz9.12lem3 9206 onssnum 9454 dfac12lem2 9558 ackbij1lem16 9645 cfslb2n 9678 hsmexlem1 9836 |
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