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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 | elex 3510 | . . 3 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ∈ V) | |
2 | 0ex 5202 | . . 3 ⊢ ∅ ∈ V | |
3 | 1, 2 | jctir 521 | . 2 ⊢ (𝐴 ∈ 𝒫 No → (𝐴 ∈ V ∧ ∅ ∈ V)) |
4 | elpwi 4547 | . . 3 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ⊆ No ) | |
5 | 0ss 4347 | . . . 4 ⊢ ∅ ⊆ No | |
6 | 5 | a1i 11 | . . 3 ⊢ (𝐴 ∈ 𝒫 No → ∅ ⊆ No ) |
7 | ral0 4452 | . . . . 5 ⊢ ∀𝑦 ∈ ∅ 𝑥 <s 𝑦 | |
8 | 7 | rgenw 3147 | . . . 4 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ∅ 𝑥 <s 𝑦 |
9 | 8 | a1i 11 | . . 3 ⊢ (𝐴 ∈ 𝒫 No → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ∅ 𝑥 <s 𝑦) |
10 | 4, 6, 9 | 3jca 1120 | . 2 ⊢ (𝐴 ∈ 𝒫 No → (𝐴 ⊆ No ∧ ∅ ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ∅ 𝑥 <s 𝑦)) |
11 | brsslt 33151 | . 2 ⊢ (𝐴 <<s ∅ ↔ ((𝐴 ∈ V ∧ ∅ ∈ V) ∧ (𝐴 ⊆ No ∧ ∅ ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ∅ 𝑥 <s 𝑦))) | |
12 | 3, 10, 11 | sylanbrc 583 | 1 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 <<s ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 ∈ wcel 2105 ∀wral 3135 Vcvv 3492 ⊆ wss 3933 ∅c0 4288 𝒫 cpw 4535 class class class wbr 5057 No csur 33044 <s cslt 33045 <<s csslt 33147 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-sslt 33148 |
This theorem is referenced by: (None) |
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