Theorem ssltsnb 27767

Description: Surreal set less-than of two singletons. (Contributed by Scott Fenton, 18-Jan-2026.)

Hypotheses

Ref	Expression
ssltsnb.1	⊢ (𝜑 → 𝐴 ∈ No )
ssltsnb.2	⊢ (𝜑 → 𝐵 ∈ No )

Assertion

Ref	Expression
ssltsnb	⊢ (𝜑 → ({𝐴} <<s {𝐵} ↔ 𝐴 <s 𝐵))

Proof of Theorem ssltsnb

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 5380	. . . 4 ⊢ {𝐴} ∈ V
2		snex 5380	. . . 4 ⊢ {𝐵} ∈ V
3	1, 2	pm3.2i 470	. . 3 ⊢ ({𝐴} ∈ V ∧ {𝐵} ∈ V)
4		brsslt 27760	. . 3 ⊢ ({𝐴} <<s {𝐵} ↔ (({𝐴} ∈ V ∧ {𝐵} ∈ V) ∧ ({𝐴} ⊆ No ∧ {𝐵} ⊆ No ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝑥 <s 𝑦)))
5	3, 4	mpbiran 710	. 2 ⊢ ({𝐴} <<s {𝐵} ↔ ({𝐴} ⊆ No ∧ {𝐵} ⊆ No ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝑥 <s 𝑦))
6		df-3an 1089	. . 3 ⊢ (({𝐴} ⊆ No ∧ {𝐵} ⊆ No ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝑥 <s 𝑦) ↔ (({𝐴} ⊆ No ∧ {𝐵} ⊆ No ) ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝑥 <s 𝑦))
7		ssltsnb.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ No )
8		breq1 5100	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 <s 𝑦 ↔ 𝐴 <s 𝑦))
9	8	ralbidv 3158	. . . . . 6 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ {𝐵}𝑥 <s 𝑦 ↔ ∀𝑦 ∈ {𝐵}𝐴 <s 𝑦))
10	9	ralsng 4631	. . . . 5 ⊢ (𝐴 ∈ No → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝑥 <s 𝑦 ↔ ∀𝑦 ∈ {𝐵}𝐴 <s 𝑦))
11	7, 10	syl 17	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝑥 <s 𝑦 ↔ ∀𝑦 ∈ {𝐵}𝐴 <s 𝑦))
12	7	snssd 4764	. . . . . 6 ⊢ (𝜑 → {𝐴} ⊆ No )
13		ssltsnb.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ No )
14	13	snssd 4764	. . . . . 6 ⊢ (𝜑 → {𝐵} ⊆ No )
15	12, 14	jca 511	. . . . 5 ⊢ (𝜑 → ({𝐴} ⊆ No ∧ {𝐵} ⊆ No ))
16	15	biantrurd 532	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝑥 <s 𝑦 ↔ (({𝐴} ⊆ No ∧ {𝐵} ⊆ No ) ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝑥 <s 𝑦)))
17		breq2 5101	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 <s 𝑦 ↔ 𝐴 <s 𝐵))
18	17	ralsng 4631	. . . . 5 ⊢ (𝐵 ∈ No → (∀𝑦 ∈ {𝐵}𝐴 <s 𝑦 ↔ 𝐴 <s 𝐵))
19	13, 18	syl 17	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ {𝐵}𝐴 <s 𝑦 ↔ 𝐴 <s 𝐵))
20	11, 16, 19	3bitr3d 309	. . 3 ⊢ (𝜑 → ((({𝐴} ⊆ No ∧ {𝐵} ⊆ No ) ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝑥 <s 𝑦) ↔ 𝐴 <s 𝐵))
21	6, 20	bitr2id 284	. 2 ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ ({𝐴} ⊆ No ∧ {𝐵} ⊆ No ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝑥 <s 𝑦)))
22	5, 21	bitr4id 290	1 ⊢ (𝜑 → ({𝐴} <<s {𝐵} ↔ 𝐴 <s 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3050  Vcvv 3439   ⊆ wss 3900  {csn 4579   class class class wbr 5097   No csur 27609   <s cslt 27610   <<s csslt 27755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-xp 5629  df-sslt 27756

This theorem is referenced by:  ssltsn  27768  pw2cut2  28439