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Mirrors > Home > MPE Home > Th. List > onelini | Structured version Visualization version GIF version |
Description: An element of an ordinal number equals the intersection with it. (Contributed by NM, 11-Jun-1994.) |
Ref | Expression |
---|---|
on.1 | ⊢ 𝐴 ∈ On |
Ref | Expression |
---|---|
onelini | ⊢ (𝐵 ∈ 𝐴 → 𝐵 = (𝐵 ∩ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | on.1 | . . 3 ⊢ 𝐴 ∈ On | |
2 | 1 | onelssi 6501 | . 2 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴) |
3 | dfss 3982 | . 2 ⊢ (𝐵 ⊆ 𝐴 ↔ 𝐵 = (𝐵 ∩ 𝐴)) | |
4 | 2, 3 | sylib 218 | 1 ⊢ (𝐵 ∈ 𝐴 → 𝐵 = (𝐵 ∩ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∩ cin 3962 ⊆ wss 3963 Oncon0 6386 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-v 3480 df-in 3970 df-ss 3980 df-uni 4913 df-tr 5266 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-on 6390 |
This theorem is referenced by: (None) |
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