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Mirrors > Home > MPE Home > Th. List > Mathboxes > nulssgt | Structured version Visualization version GIF version |
Description: The empty set is greater than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.) |
Ref | Expression |
---|---|
nulssgt | ⊢ (𝐴 ∈ 𝒫 No → 𝐴 <<s ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ∈ 𝒫 No ) | |
2 | 0ex 5185 | . . 3 ⊢ ∅ ∈ V | |
3 | 2 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ ∈ V) |
4 | elpwi 4508 | . 2 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ⊆ No ) | |
5 | 0ss 4297 | . . 3 ⊢ ∅ ⊆ No | |
6 | 5 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ ⊆ No ) |
7 | noel 4231 | . . . 4 ⊢ ¬ 𝑦 ∈ ∅ | |
8 | 7 | pm2.21i 119 | . . 3 ⊢ (𝑦 ∈ ∅ → 𝑥 <s 𝑦) |
9 | 8 | 3ad2ant3 1137 | . 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ∅) → 𝑥 <s 𝑦) |
10 | 1, 3, 4, 6, 9 | ssltd 33672 | 1 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 <<s ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2112 Vcvv 3398 ⊆ wss 3853 ∅c0 4223 𝒫 cpw 4499 class class class wbr 5039 No csur 33529 <s cslt 33530 <<s csslt 33661 |
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-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-xp 5542 df-sslt 33662 |
This theorem is referenced by: 0sno 33706 1sno 33707 bday0s 33708 0slt1s 33709 bday0b 33710 bday1s 33711 made0 33743 |
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