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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 4892 | . . . . . 6 ⊢ (𝐴 ⊆ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥) | |
2 | ssel 3961 | . . . . . . . . . . . 12 ⊢ (𝐵 ⊆ On → (𝐴 ∈ 𝐵 → 𝐴 ∈ On)) | |
3 | ssel 3961 | . . . . . . . . . . . 12 ⊢ (𝐵 ⊆ On → (𝑥 ∈ 𝐵 → 𝑥 ∈ On)) | |
4 | 2, 3 | anim12d 610 | . . . . . . . . . . 11 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝐴 ∈ On ∧ 𝑥 ∈ On))) |
5 | ontri1 6225 | . . . . . . . . . . 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 455 | . . . . . . . . 9 ⊢ ((𝐵 ⊆ On ∧ 𝐴 ∈ 𝐵) → (𝑥 ∈ 𝐵 → (𝐴 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐴))) |
8 | 7 | pm5.74d 275 | . . . . . . . 8 ⊢ ((𝐵 ⊆ On ∧ 𝐴 ∈ 𝐵) → ((𝑥 ∈ 𝐵 → 𝐴 ⊆ 𝑥) ↔ (𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴))) |
9 | con2b 362 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)) | |
10 | 8, 9 | syl6bb 289 | . . . . . . 7 ⊢ ((𝐵 ⊆ On ∧ 𝐴 ∈ 𝐵) → ((𝑥 ∈ 𝐵 → 𝐴 ⊆ 𝑥) ↔ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))) |
11 | 10 | ralbidv2 3195 | . . . . . 6 ⊢ ((𝐵 ⊆ On ∧ 𝐴 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)) |
12 | 1, 11 | syl5bb 285 | . . . . 5 ⊢ ((𝐵 ⊆ On ∧ 𝐴 ∈ 𝐵) → (𝐴 ⊆ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)) |
13 | 12 | biimprd 250 | . . . 4 ⊢ ((𝐵 ⊆ On ∧ 𝐴 ∈ 𝐵) → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 → 𝐴 ⊆ ∩ 𝐵)) |
14 | 13 | expimpd 456 | . . 3 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → 𝐴 ⊆ ∩ 𝐵)) |
15 | intss1 4891 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 → ∩ 𝐵 ⊆ 𝐴) | |
16 | 15 | a1i 11 | . . . 4 ⊢ (𝐵 ⊆ On → (𝐴 ∈ 𝐵 → ∩ 𝐵 ⊆ 𝐴)) |
17 | 16 | adantrd 494 | . . 3 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → ∩ 𝐵 ⊆ 𝐴)) |
18 | 14, 17 | jcad 515 | . 2 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → (𝐴 ⊆ ∩ 𝐵 ∧ ∩ 𝐵 ⊆ 𝐴))) |
19 | eqss 3982 | . 2 ⊢ (𝐴 = ∩ 𝐵 ↔ (𝐴 ⊆ ∩ 𝐵 ∧ ∩ 𝐵 ⊆ 𝐴)) | |
20 | 18, 19 | syl6ibr 254 | 1 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → 𝐴 = ∩ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ⊆ wss 3936 ∩ cint 4876 Oncon0 6191 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-int 4877 df-br 5067 df-opab 5129 df-tr 5173 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-ord 6194 df-on 6195 |
This theorem is referenced by: oneqmin 7520 alephval3 9536 cfsuc 9679 alephval2 9994 |
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