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Mirrors > Home > MPE Home > Th. List > Mathboxes > brsslt | Structured version Visualization version GIF version |
Description: Binary relation form of the surreal set less-than relation. (Contributed by Scott Fenton, 8-Dec-2021.) |
Ref | Expression |
---|---|
brsslt | ⊢ (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sslt 33148 | . . 3 ⊢ <<s = {〈𝑎, 𝑏〉 ∣ (𝑎 ⊆ No ∧ 𝑏 ⊆ No ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦)} | |
2 | 1 | bropaex12 5635 | . 2 ⊢ (𝐴 <<s 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
3 | sseq1 3989 | . . . 4 ⊢ (𝑎 = 𝐴 → (𝑎 ⊆ No ↔ 𝐴 ⊆ No )) | |
4 | raleq 3403 | . . . 4 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦)) | |
5 | 3, 4 | 3anbi13d 1429 | . . 3 ⊢ (𝑎 = 𝐴 → ((𝑎 ⊆ No ∧ 𝑏 ⊆ No ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦) ↔ (𝐴 ⊆ No ∧ 𝑏 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦))) |
6 | sseq1 3989 | . . . 4 ⊢ (𝑏 = 𝐵 → (𝑏 ⊆ No ↔ 𝐵 ⊆ No )) | |
7 | raleq 3403 | . . . . 5 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ 𝑏 𝑥 <s 𝑦 ↔ ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)) | |
8 | 7 | ralbidv 3194 | . . . 4 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)) |
9 | 6, 8 | 3anbi23d 1430 | . . 3 ⊢ (𝑏 = 𝐵 → ((𝐴 ⊆ No ∧ 𝑏 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦) ↔ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦))) |
10 | 5, 9, 1 | brabg 5417 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 <<s 𝐵 ↔ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦))) |
11 | 2, 10 | biadanii 818 | 1 ⊢ (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 Vcvv 3492 ⊆ wss 3933 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-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: ssltex1 33152 ssltex2 33153 ssltss1 33154 ssltss2 33155 ssltsep 33156 sssslt1 33157 sssslt2 33158 nulsslt 33159 nulssgt 33160 conway 33161 sslttr 33165 ssltun1 33166 ssltun2 33167 etasslt 33171 slerec 33174 |
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