![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 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 5265 | . . 3 ⊢ ∅ ∈ V | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ ∈ V) |
3 | id 22 | . 2 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ∈ 𝒫 No ) | |
4 | 0ss 4357 | . . 3 ⊢ ∅ ⊆ No | |
5 | 4 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ ⊆ No ) |
6 | elpwi 4568 | . 2 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ⊆ No ) | |
7 | noel 4291 | . . . 4 ⊢ ¬ 𝑥 ∈ ∅ | |
8 | 7 | pm2.21i 119 | . . 3 ⊢ (𝑥 ∈ ∅ → 𝑥 <s 𝑦) |
9 | 8 | 3ad2ant2 1135 | . 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ 𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴) → 𝑥 <s 𝑦) |
10 | 2, 3, 5, 6, 9 | ssltd 27153 | 1 ⊢ (𝐴 ∈ 𝒫 No → ∅ <<s 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 Vcvv 3444 ⊆ wss 3911 ∅c0 4283 𝒫 cpw 4561 class class class wbr 5106 No csur 27004 <s cslt 27005 <<s csslt 27142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-xp 5640 df-sslt 27143 |
This theorem is referenced by: bday0s 27189 bday0b 27191 |
Copyright terms: Public domain | W3C validator |