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| 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 5436 | . . 3 ⊢ {𝐴} ∈ V | |
| 2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → {𝐴} ∈ V) | 
| 3 | snex 5436 | . . 3 ⊢ {𝐵} ∈ V | |
| 4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → {𝐵} ∈ V) | 
| 5 | ssltsn.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ No ) | |
| 6 | 5 | snssd 4809 | . 2 ⊢ (𝜑 → {𝐴} ⊆ No ) | 
| 7 | ssltsn.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ No ) | |
| 8 | 7 | snssd 4809 | . 2 ⊢ (𝜑 → {𝐵} ⊆ No ) | 
| 9 | ssltsn.3 | . . . 4 ⊢ (𝜑 → 𝐴 <s 𝐵) | |
| 10 | velsn 4642 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
| 11 | velsn 4642 | . . . . 5 ⊢ (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵) | |
| 12 | breq12 5148 | . . . . 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 1117 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐵}) → 𝑥 <s 𝑦) | 
| 16 | 2, 4, 6, 8, 15 | ssltd 27836 | 1 ⊢ (𝜑 → {𝐴} <<s {𝐵}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 class class class wbr 5143 No csur 27684 <s cslt 27685 <<s csslt 27825 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-sslt 27826 | 
| This theorem is referenced by: n0scut 28338 zscut 28393 halfcut 28416 pw2bday 28418 addhalfcut 28419 zs12bday 28424 | 
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