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Mirrors > Home > MPE Home > Th. List > Mathboxes > supubt | Structured version Visualization version GIF version |
Description: Upper bound property of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
supubt | ⊢ ((𝑅 Or 𝐴 ∧ ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) → (𝐶 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 482 | . 2 ⊢ ((𝑅 Or 𝐴 ∧ ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) → 𝑅 Or 𝐴) | |
2 | simpr 484 | . 2 ⊢ ((𝑅 Or 𝐴 ∧ ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) | |
3 | 1, 2 | supub 9496 | 1 ⊢ ((𝑅 Or 𝐴 ∧ ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) → (𝐶 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∈ wcel 2105 ∀wral 3058 ∃wrex 3067 class class class wbr 5147 Or wor 5595 supcsup 9477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-po 5596 df-so 5597 df-iota 6515 df-riota 7387 df-sup 9479 |
This theorem is referenced by: (None) |
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