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Mirrors > Home > MPE Home > Th. List > Mathboxes > nulsslt | Structured version Visualization version GIF version |
Description: The empty set is less than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.) |
Ref | Expression |
---|---|
nulsslt | ⊢ (𝐴 ∈ 𝒫 No → ∅ <<s 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5226 | . . 3 ⊢ ∅ ∈ V | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ ∈ V) |
3 | id 22 | . 2 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ∈ 𝒫 No ) | |
4 | 0ss 4327 | . . 3 ⊢ ∅ ⊆ No | |
5 | 4 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ ⊆ No ) |
6 | elpwi 4539 | . 2 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ⊆ No ) | |
7 | noel 4261 | . . . 4 ⊢ ¬ 𝑥 ∈ ∅ | |
8 | 7 | pm2.21i 119 | . . 3 ⊢ (𝑥 ∈ ∅ → 𝑥 <s 𝑦) |
9 | 8 | 3ad2ant2 1132 | . 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ 𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴) → 𝑥 <s 𝑦) |
10 | 2, 3, 5, 6, 9 | ssltd 33913 | 1 ⊢ (𝐴 ∈ 𝒫 No → ∅ <<s 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2108 Vcvv 3422 ⊆ wss 3883 ∅c0 4253 𝒫 cpw 4530 class class class wbr 5070 No csur 33770 <s cslt 33771 <<s csslt 33902 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pr 5347 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2817 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-br 5071 df-opab 5133 df-xp 5586 df-sslt 33903 |
This theorem is referenced by: bday0s 33949 bday0b 33951 |
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