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Mirrors > Home > MPE Home > Th. List > onintss | Structured version Visualization version GIF version |
Description: If a property is true for an ordinal number, then the minimum ordinal number for which it is true is smaller or equal. Theorem Schema 61 of [Suppes] p. 228. (Contributed by NM, 3-Oct-2003.) |
Ref | Expression |
---|---|
onintss.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
onintss | ⊢ (𝐴 ∈ On → (𝜓 → ∩ {𝑥 ∈ On ∣ 𝜑} ⊆ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onintss.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
2 | 1 | intminss 4864 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝜓) → ∩ {𝑥 ∈ On ∣ 𝜑} ⊆ 𝐴) |
3 | 2 | ex 416 | 1 ⊢ (𝐴 ∈ On → (𝜓 → ∩ {𝑥 ∈ On ∣ 𝜑} ⊆ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 {crab 3110 ⊆ wss 3881 ∩ cint 4838 Oncon0 6159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-in 3888 df-ss 3898 df-int 4839 |
This theorem is referenced by: rankval3b 9239 cardne 9378 noextenddif 33288 |
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