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Mirrors > Home > MPE Home > Th. List > onsseleq | Structured version Visualization version GIF version |
Description: Relationship between subset and membership of an ordinal number. (Contributed by NM, 15-Sep-1995.) |
Ref | Expression |
---|---|
onsseleq | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 5977 | . 2 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
2 | eloni 5977 | . 2 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
3 | ordsseleq 5996 | . 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) | |
4 | 1, 2, 3 | syl2an 589 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∨ wo 878 = wceq 1656 ∈ wcel 2164 ⊆ wss 3798 Ord word 5966 Oncon0 5967 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-tr 4978 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-ord 5970 df-on 5971 |
This theorem is referenced by: onsseli 6081 on0eqel 6084 onmindif2 7278 omword 7922 oeword 7942 oewordi 7943 dffi3 8612 cantnflem1d 8869 cantnflem1 8870 r1ord3g 8926 alephdom 9224 cardaleph 9232 cfsmolem 9414 ttukeylem5 9657 alephreg 9726 inar1 9919 gruina 9962 om2uzlt2i 13052 nolt02o 32379 nosupbnd2lem1 32395 |
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