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Mirrors > Home > MPE Home > Th. List > supcl | Structured version Visualization version GIF version |
Description: A supremum belongs to its base class (closure law). See also supub 9053 and suplub 9054. (Contributed by NM, 12-Oct-2004.) |
Ref | Expression |
---|---|
supmo.1 | ⊢ (𝜑 → 𝑅 Or 𝐴) |
supcl.2 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) |
Ref | Expression |
---|---|
supcl | ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | supmo.1 | . . 3 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
2 | 1 | supval2 9049 | . 2 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))) |
3 | supcl.2 | . . . 4 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) | |
4 | 1, 3 | supeu 9048 | . . 3 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) |
5 | riotacl 7166 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ∈ 𝐴) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ∈ 𝐴) |
7 | 2, 6 | eqeltrd 2831 | 1 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∈ wcel 2112 ∀wral 3051 ∃wrex 3052 ∃!wreu 3053 class class class wbr 5039 Or wor 5452 ℩crio 7147 supcsup 9034 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-po 5453 df-so 5454 df-iota 6316 df-riota 7148 df-sup 9036 |
This theorem is referenced by: suplub2 9055 supiso 9069 infcl 9082 inflb 9083 infglb 9084 infglbb 9085 suprcl 11757 supxrcl 12870 dgrcl 25081 supssd 30718 xrsupssd 30756 esum2d 31727 oddpwdc 31987 supclt 35582 |
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