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Mirrors > Home > MPE Home > Th. List > oneqmini | Structured version Visualization version GIF version |
Description: A way to show that an ordinal number equals the minimum of a collection of ordinal numbers: it must be in the collection, and it must not be larger than any member of the collection. (Contributed by NM, 14-Nov-2003.) |
Ref | Expression |
---|---|
oneqmini | ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → 𝐴 = ∩ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssint 4913 | . . . . . 6 ⊢ (𝐴 ⊆ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥) | |
2 | ssel 3925 | . . . . . . . . . . . 12 ⊢ (𝐵 ⊆ On → (𝐴 ∈ 𝐵 → 𝐴 ∈ On)) | |
3 | ssel 3925 | . . . . . . . . . . . 12 ⊢ (𝐵 ⊆ On → (𝑥 ∈ 𝐵 → 𝑥 ∈ On)) | |
4 | 2, 3 | anim12d 609 | . . . . . . . . . . 11 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝐴 ∈ On ∧ 𝑥 ∈ On))) |
5 | ontri1 6337 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐴)) | |
6 | 4, 5 | syl6 35 | . . . . . . . . . 10 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝐴 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐴))) |
7 | 6 | expdimp 453 | . . . . . . . . 9 ⊢ ((𝐵 ⊆ On ∧ 𝐴 ∈ 𝐵) → (𝑥 ∈ 𝐵 → (𝐴 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐴))) |
8 | 7 | pm5.74d 272 | . . . . . . . 8 ⊢ ((𝐵 ⊆ On ∧ 𝐴 ∈ 𝐵) → ((𝑥 ∈ 𝐵 → 𝐴 ⊆ 𝑥) ↔ (𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴))) |
9 | con2b 359 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)) | |
10 | 8, 9 | bitrdi 286 | . . . . . . 7 ⊢ ((𝐵 ⊆ On ∧ 𝐴 ∈ 𝐵) → ((𝑥 ∈ 𝐵 → 𝐴 ⊆ 𝑥) ↔ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))) |
11 | 10 | ralbidv2 3166 | . . . . . 6 ⊢ ((𝐵 ⊆ On ∧ 𝐴 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)) |
12 | 1, 11 | bitrid 282 | . . . . 5 ⊢ ((𝐵 ⊆ On ∧ 𝐴 ∈ 𝐵) → (𝐴 ⊆ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)) |
13 | 12 | biimprd 247 | . . . 4 ⊢ ((𝐵 ⊆ On ∧ 𝐴 ∈ 𝐵) → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 → 𝐴 ⊆ ∩ 𝐵)) |
14 | 13 | expimpd 454 | . . 3 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → 𝐴 ⊆ ∩ 𝐵)) |
15 | intss1 4912 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 → ∩ 𝐵 ⊆ 𝐴) | |
16 | 15 | a1i 11 | . . . 4 ⊢ (𝐵 ⊆ On → (𝐴 ∈ 𝐵 → ∩ 𝐵 ⊆ 𝐴)) |
17 | 16 | adantrd 492 | . . 3 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → ∩ 𝐵 ⊆ 𝐴)) |
18 | 14, 17 | jcad 513 | . 2 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → (𝐴 ⊆ ∩ 𝐵 ∧ ∩ 𝐵 ⊆ 𝐴))) |
19 | eqss 3947 | . 2 ⊢ (𝐴 = ∩ 𝐵 ↔ (𝐴 ⊆ ∩ 𝐵 ∧ ∩ 𝐵 ⊆ 𝐴)) | |
20 | 18, 19 | syl6ibr 251 | 1 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → 𝐴 = ∩ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ∀wral 3061 ⊆ wss 3898 ∩ cint 4895 Oncon0 6303 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-11 2153 ax-ext 2707 ax-sep 5244 ax-nul 5251 ax-pr 5373 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4271 df-if 4475 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4854 df-int 4896 df-br 5094 df-opab 5156 df-tr 5211 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5576 df-we 5578 df-ord 6306 df-on 6307 |
This theorem is referenced by: oneqmin 7714 alephval3 9968 cfsuc 10115 alephval2 10430 |
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