Theorem sltssnb 27839

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

Hypotheses

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

Assertion

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

Proof of Theorem sltssnb

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

Step	Hyp	Ref	Expression
1		snex 5395	. . . 4 ⊢ {𝐴} ∈ V
2		snex 5395	. . . 4 ⊢ {𝐵} ∈ V
3	1, 2	pm3.2i 474	. . 3 ⊢ ({𝐴} ∈ V ∧ {𝐵} ∈ V)
4		brslts 27832	. . 3 ⊢ ({𝐴} <<s {𝐵} ↔ (({𝐴} ∈ V ∧ {𝐵} ∈ V) ∧ ({𝐴} ⊆ No ∧ {𝐵} ⊆ No ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝑥 <s 𝑦)))
5	3, 4	mpbiran 719	. 2 ⊢ ({𝐴} <<s {𝐵} ↔ ({𝐴} ⊆ No ∧ {𝐵} ⊆ No ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝑥 <s 𝑦))
6		df-3an 1099	. . 3 ⊢ (({𝐴} ⊆ No ∧ {𝐵} ⊆ No ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝑥 <s 𝑦) ↔ (({𝐴} ⊆ No ∧ {𝐵} ⊆ No ) ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝑥 <s 𝑦))
7		sltssnb.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ No )
8		breq1 5102	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 <s 𝑦 ↔ 𝐴 <s 𝑦))
9	8	ralbidv 3184	. . . . . 6 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ {𝐵}𝑥 <s 𝑦 ↔ ∀𝑦 ∈ {𝐵}𝐴 <s 𝑦))
10	9	ralsng 4633	. . . . 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 4744	. . . . . 6 ⊢ (𝜑 → {𝐴} ⊆ No )
13		sltssnb.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ No )
14	13	snssd 4744	. . . . . 6 ⊢ (𝜑 → {𝐵} ⊆ No )
15	12, 14	jca 519	. . . . 5 ⊢ (𝜑 → ({𝐴} ⊆ No ∧ {𝐵} ⊆ No ))
16	15	biantrurd 540	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝑥 <s 𝑦 ↔ (({𝐴} ⊆ No ∧ {𝐵} ⊆ No ) ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝑥 <s 𝑦)))
17		breq2 5103	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 <s 𝑦 ↔ 𝐴 <s 𝐵))
18	17	ralsng 4633	. . . . 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 311	. . 3 ⊢ (𝜑 → ((({𝐴} ⊆ No ∧ {𝐵} ⊆ No ) ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝑥 <s 𝑦) ↔ 𝐴 <s 𝐵))
21	6, 20	bitr2id 286	. 2 ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ ({𝐴} ⊆ No ∧ {𝐵} ⊆ No ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝑥 <s 𝑦)))
22	5, 21	bitr4id 292	1 ⊢ (𝜑 → ({𝐴} <<s {𝐵} ↔ 𝐴 <s 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075  Vcvv 3453   ⊆ wss 3904  {csn 4581   class class class wbr 5099   No csur 27681   <s clts 27682   <<s cslts 27827

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5651  df-slts 27828

This theorem is referenced by:  sltssn  27840  pw2cut2  28532