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Mirrors > Home > MPE Home > Th. List > Mathboxes > supex2g | Structured version Visualization version GIF version |
Description: Existence of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
supex2g | ⊢ (𝐴 ∈ 𝐶 → sup(𝐵, 𝐴, 𝑅) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sup 8900 | . 2 ⊢ sup(𝐵, 𝐴, 𝑅) = ∪ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} | |
2 | rabexg 5227 | . . 3 ⊢ (𝐴 ∈ 𝐶 → {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} ∈ V) | |
3 | 2 | uniexd 7462 | . 2 ⊢ (𝐴 ∈ 𝐶 → ∪ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} ∈ V) |
4 | 1, 3 | eqeltrid 2917 | 1 ⊢ (𝐴 ∈ 𝐶 → sup(𝐵, 𝐴, 𝑅) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 {crab 3142 Vcvv 3495 ∪ cuni 4832 class class class wbr 5059 supcsup 8898 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-rab 3147 df-v 3497 df-in 3943 df-ss 3952 df-uni 4833 df-sup 8900 |
This theorem is referenced by: (None) |
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