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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 5986 | . 2 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
2 | eloni 5986 | . 2 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
3 | ordsseleq 6005 | . 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 836 = wceq 1601 ∈ wcel 2106 ⊆ wss 3791 Ord word 5975 Oncon0 5976 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-tr 4988 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-ord 5979 df-on 5980 |
This theorem is referenced by: onsseli 6090 on0eqel 6093 onmindif2 7290 omword 7934 oeword 7954 oewordi 7955 dffi3 8625 cantnflem1d 8882 cantnflem1 8883 r1ord3g 8939 alephdom 9237 cardaleph 9245 cfsmolem 9427 ttukeylem5 9670 alephreg 9739 inar1 9932 gruina 9975 om2uzlt2i 13069 nolt02o 32434 nosupbnd2lem1 32450 |
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