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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 5974 | . 2 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
2 | eloni 5974 | . 2 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
3 | ordelssne 5991 | . 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵))) | |
4 | 1, 2, 3 | syl2an 591 | 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 ∈ wcel 2166 ≠ wne 3000 ⊆ wss 3799 Ord word 5963 Oncon0 5964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-tr 4977 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-ord 5967 df-on 5968 |
This theorem is referenced by: tfindsg 7322 findsg 7355 oancom 8826 cardsdom2 9128 alephord 9212 scutbdaylt 32462 |
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