![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > onnminsb | Structured version Visualization version GIF version |
Description: An ordinal number smaller than the minimum of a set of ordinal numbers does not have the property determining that set. 𝜓 is the wff resulting from the substitution of 𝐴 for 𝑥 in wff 𝜑. (Contributed by NM, 9-Nov-2003.) |
Ref | Expression |
---|---|
onnminsb.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
onnminsb | ⊢ (𝐴 ∈ On → (𝐴 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} → ¬ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onnminsb.1 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
2 | 1 | elrab 3650 | . . . 4 ⊢ (𝐴 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝐴 ∈ On ∧ 𝜓)) |
3 | ssrab2 4042 | . . . . 5 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On | |
4 | onnmin 7738 | . . . . 5 ⊢ (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ 𝐴 ∈ {𝑥 ∈ On ∣ 𝜑}) → ¬ 𝐴 ∈ ∩ {𝑥 ∈ On ∣ 𝜑}) | |
5 | 3, 4 | mpan 689 | . . . 4 ⊢ (𝐴 ∈ {𝑥 ∈ On ∣ 𝜑} → ¬ 𝐴 ∈ ∩ {𝑥 ∈ On ∣ 𝜑}) |
6 | 2, 5 | sylbir 234 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝜓) → ¬ 𝐴 ∈ ∩ {𝑥 ∈ On ∣ 𝜑}) |
7 | 6 | ex 414 | . 2 ⊢ (𝐴 ∈ On → (𝜓 → ¬ 𝐴 ∈ ∩ {𝑥 ∈ On ∣ 𝜑})) |
8 | 7 | con2d 134 | 1 ⊢ (𝐴 ∈ On → (𝐴 ∈ ∩ {𝑥 ∈ On ∣ 𝜑} → ¬ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {crab 3410 ⊆ wss 3915 ∩ cint 4912 Oncon0 6322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-br 5111 df-opab 5173 df-tr 5228 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-ord 6325 df-on 6326 |
This theorem is referenced by: onminex 7742 oawordeulem 8506 oeeulem 8553 nnawordex 8589 tcrank 9827 alephnbtwn 10014 cardaleph 10032 cardmin 10507 sltval2 27020 nosepeq 27049 nosupbnd2lem1 27079 noinfbnd2lem1 27094 naddwordnexlem4 41747 |
Copyright terms: Public domain | W3C validator |