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Mirrors > Home > MPE Home > Th. List > ssltsn | Structured version Visualization version GIF version |
Description: Surreal set less-than of two singletons. (Contributed by Scott Fenton, 17-Mar-2025.) |
Ref | Expression |
---|---|
ssltsn.1 | ⊢ (𝜑 → 𝐴 ∈ No ) |
ssltsn.2 | ⊢ (𝜑 → 𝐵 ∈ No ) |
ssltsn.3 | ⊢ (𝜑 → 𝐴 <s 𝐵) |
Ref | Expression |
---|---|
ssltsn | ⊢ (𝜑 → {𝐴} <<s {𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex 5442 | . . 3 ⊢ {𝐴} ∈ V | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → {𝐴} ∈ V) |
3 | snex 5442 | . . 3 ⊢ {𝐵} ∈ V | |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → {𝐵} ∈ V) |
5 | ssltsn.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ No ) | |
6 | 5 | snssd 4814 | . 2 ⊢ (𝜑 → {𝐴} ⊆ No ) |
7 | ssltsn.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ No ) | |
8 | 7 | snssd 4814 | . 2 ⊢ (𝜑 → {𝐵} ⊆ No ) |
9 | ssltsn.3 | . . . 4 ⊢ (𝜑 → 𝐴 <s 𝐵) | |
10 | velsn 4647 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
11 | velsn 4647 | . . . . 5 ⊢ (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵) | |
12 | breq12 5153 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 <s 𝑦 ↔ 𝐴 <s 𝐵)) | |
13 | 10, 11, 12 | syl2anb 598 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐵}) → (𝑥 <s 𝑦 ↔ 𝐴 <s 𝐵)) |
14 | 9, 13 | syl5ibrcom 247 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐵}) → 𝑥 <s 𝑦)) |
15 | 14 | 3impib 1115 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐵}) → 𝑥 <s 𝑦) |
16 | 2, 4, 6, 8, 15 | ssltd 27851 | 1 ⊢ (𝜑 → {𝐴} <<s {𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 {csn 4631 class class class wbr 5148 No csur 27699 <s cslt 27700 <<s csslt 27840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-sslt 27841 |
This theorem is referenced by: n0scut 28353 zscut 28408 halfcut 28431 pw2bday 28433 addhalfcut 28434 zs12bday 28439 |
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