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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 6381 | . 2 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
2 | eloni 6381 | . 2 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
3 | ordsseleq 6400 | . 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) | |
4 | 1, 2, 3 | syl2an 594 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ⊆ wss 3944 Ord word 6370 Oncon0 6371 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-tr 5267 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-ord 6374 df-on 6375 |
This theorem is referenced by: onsseli 6492 on0eqel 6495 onmindif2 7811 omword 8591 oeword 8611 oewordi 8612 dffi3 9456 cantnflem1d 9713 cantnflem1 9714 r1ord3g 9804 alephdom 10106 cardaleph 10114 cfsmolem 10295 ttukeylem5 10538 alephreg 10607 inar1 10800 gruina 10843 om2uzlt2i 13952 nolt02o 27674 nogt01o 27675 nosupbnd2lem1 27694 noinfbnd2lem1 27709 madebday 27872 om2noseqlt2 28223 oege2 42878 ontric3g 43094 |
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