Theorem sltssnb 27786

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 5375	. . . 4 ⊢ {𝐴} ∈ V
2		snex 5375	. . . 4 ⊢ {𝐵} ∈ V
3	1, 2	pm3.2i 471	. . 3 ⊢ ({𝐴} ∈ V ∧ {𝐵} ∈ V)
4		brslts 27779	. . 3 ⊢ ({𝐴} <<s {𝐵} ↔ (({𝐴} ∈ V ∧ {𝐵} ∈ V) ∧ ({𝐴} ⊆ No ∧ {𝐵} ⊆ No ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝑥 <s 𝑦)))
5	3, 4	mpbiran 715	. 2 ⊢ ({𝐴} <<s {𝐵} ↔ ({𝐴} ⊆ No ∧ {𝐵} ⊆ No ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝑥 <s 𝑦))
6		df-3an 1094	. . 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 5082	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 <s 𝑦 ↔ 𝐴 <s 𝑦))
9	8	ralbidv 3163	. . . . . 6 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ {𝐵}𝑥 <s 𝑦 ↔ ∀𝑦 ∈ {𝐵}𝐴 <s 𝑦))
10	9	ralsng 4614	. . . . 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 4725	. . . . . 6 ⊢ (𝜑 → {𝐴} ⊆ No )
13		sltssnb.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ No )
14	13	snssd 4725	. . . . . 6 ⊢ (𝜑 → {𝐵} ⊆ No )
15	12, 14	jca 516	. . . . 5 ⊢ (𝜑 → ({𝐴} ⊆ No ∧ {𝐵} ⊆ No ))
16	15	biantrurd 537	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝑥 <s 𝑦 ↔ (({𝐴} ⊆ No ∧ {𝐵} ⊆ No ) ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝑥 <s 𝑦)))
17		breq2 5083	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 <s 𝑦 ↔ 𝐴 <s 𝐵))
18	17	ralsng 4614	. . . . 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 310	. . 3 ⊢ (𝜑 → ((({𝐴} ⊆ No ∧ {𝐵} ⊆ No ) ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝑥 <s 𝑦) ↔ 𝐴 <s 𝐵))
21	6, 20	bitr2id 285	. 2 ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ ({𝐴} ⊆ No ∧ {𝐵} ⊆ No ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝑥 <s 𝑦)))
22	5, 21	bitr4id 291	1 ⊢ (𝜑 → ({𝐴} <<s {𝐵} ↔ 𝐴 <s 𝐵))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3054  Vcvv 3432   ⊆ wss 3890  {csn 4562   class class class wbr 5079   No csur 27628   <s clts 27629   <<s cslts 27774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-xp 5631  df-slts 27775

This theorem is referenced by:  sltssn  27787  pw2cut2  28479