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Mirrors > Home > MPE Home > Th. List > onelpss | Structured version Visualization version GIF version |
Description: Relationship between membership and proper subset of an ordinal number. (Contributed by NM, 15-Sep-1995.) |
Ref | Expression |
---|---|
onelpss | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 6196 | . 2 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
2 | eloni 6196 | . 2 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
3 | ordelssne 6213 | . 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵))) | |
4 | 1, 2, 3 | syl2an 597 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2110 ≠ wne 3016 ⊆ wss 3936 Ord word 6185 Oncon0 6186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-tr 5166 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-ord 6189 df-on 6190 |
This theorem is referenced by: tfindsg 7569 findsg 7603 oancom 9108 cardsdom2 9411 alephord 9495 scutbdaylt 33271 |
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