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Mirrors > Home > MPE Home > Th. List > ssltd | Structured version Visualization version GIF version |
Description: Deduce surreal set less-than. (Contributed by Scott Fenton, 24-Sep-2024.) |
Ref | Expression |
---|---|
ssltd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
ssltd.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
ssltd.3 | ⊢ (𝜑 → 𝐴 ⊆ No ) |
ssltd.4 | ⊢ (𝜑 → 𝐵 ⊆ No ) |
ssltd.5 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 <s 𝑦) |
Ref | Expression |
---|---|
ssltd | ⊢ (𝜑 → 𝐴 <<s 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssltd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | 1 | elexd 3464 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) |
3 | ssltd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
4 | 3 | elexd 3464 | . 2 ⊢ (𝜑 → 𝐵 ∈ V) |
5 | ssltd.3 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ No ) | |
6 | ssltd.4 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ No ) | |
7 | ssltd.5 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 <s 𝑦) | |
8 | 7 | 3expb 1121 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑥 <s 𝑦) |
9 | 8 | ralrimivva 3194 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) |
10 | 5, 6, 9 | 3jca 1129 | . 2 ⊢ (𝜑 → (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)) |
11 | brsslt 27147 | . 2 ⊢ (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦))) | |
12 | 2, 4, 10, 11 | syl21anbrc 1345 | 1 ⊢ (𝜑 → 𝐴 <<s 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 ∈ wcel 2107 ∀wral 3061 Vcvv 3444 ⊆ wss 3911 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-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: nulsslt 27158 nulssgt 27159 sslttr 27168 ssltun1 27169 ssltun2 27170 ssltleft 27222 ssltright 27223 cofsslt 27259 coinitsslt 27260 cofcutr 27265 addsproplem2 27304 addsunif 27332 negsproplem2 27349 negsid 27361 negsunif 27372 mulsproplem10 27410 |
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