Theorem ssltsnb 27732

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 5372	. . . 4 ⊢ {𝐴} ∈ V
2		snex 5372	. . . 4 ⊢ {𝐵} ∈ V
3	1, 2	pm3.2i 470	. . 3 ⊢ ({𝐴} ∈ V ∧ {𝐵} ∈ V)
4		brsslt 27725	. . 3 ⊢ ({𝐴} <<s {𝐵} ↔ (({𝐴} ∈ V ∧ {𝐵} ∈ V) ∧ ({𝐴} ⊆ No ∧ {𝐵} ⊆ No ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝑥 <s 𝑦)))
5	3, 4	mpbiran 709	. 2 ⊢ ({𝐴} <<s {𝐵} ↔ ({𝐴} ⊆ No ∧ {𝐵} ⊆ No ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝑥 <s 𝑦))
6		df-3an 1088	. . 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 5092	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 <s 𝑦 ↔ 𝐴 <s 𝑦))
9	8	ralbidv 3155	. . . . . 6 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ {𝐵}𝑥 <s 𝑦 ↔ ∀𝑦 ∈ {𝐵}𝐴 <s 𝑦))
10	9	ralsng 4625	. . . . 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 4758	. . . . . 6 ⊢ (𝜑 → {𝐴} ⊆ No )
13		ssltsnb.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ No )
14	13	snssd 4758	. . . . . 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 5093	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 <s 𝑦 ↔ 𝐴 <s 𝐵))
18	17	ralsng 4625	. . . . 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 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  Vcvv 3436   ⊆ wss 3897  {csn 4573   class class class wbr 5089   No csur 27578   <s cslt 27579   <<s csslt 27720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-sslt 27721

This theorem is referenced by:  ssltsn  27733  pw2cut2  28382