![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > onssmin | Structured version Visualization version GIF version |
Description: A nonempty class of ordinal numbers has the smallest member. Exercise 9 of [TakeutiZaring] p. 40. (Contributed by NM, 3-Oct-2003.) |
Ref | Expression |
---|---|
onssmin | ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onint 7229 | . 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ 𝐴) | |
2 | intss1 4682 | . . 3 ⊢ (𝑦 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑦) | |
3 | 2 | rgen 3103 | . 2 ⊢ ∀𝑦 ∈ 𝐴 ∩ 𝐴 ⊆ 𝑦 |
4 | sseq1 3822 | . . . 4 ⊢ (𝑥 = ∩ 𝐴 → (𝑥 ⊆ 𝑦 ↔ ∩ 𝐴 ⊆ 𝑦)) | |
5 | 4 | ralbidv 3167 | . . 3 ⊢ (𝑥 = ∩ 𝐴 → (∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦 ↔ ∀𝑦 ∈ 𝐴 ∩ 𝐴 ⊆ 𝑦)) |
6 | 5 | rspcev 3497 | . 2 ⊢ ((∩ 𝐴 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ∩ 𝐴 ⊆ 𝑦) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) |
7 | 1, 3, 6 | sylancl 581 | 1 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ≠ wne 2971 ∀wral 3089 ∃wrex 3090 ⊆ wss 3769 ∅c0 4115 ∩ cint 4667 Oncon0 5941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-br 4844 df-opab 4906 df-tr 4946 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-ord 5944 df-on 5945 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |