Theorem sltrec 27883

Description: A comparison law for surreals considered as cuts of sets of surreals. (Contributed by Scott Fenton, 11-Dec-2021.)

Assertion

Ref	Expression
sltrec	⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑋 <s 𝑌 ↔ (∃𝑐 ∈ 𝐶 𝑋 ≤s 𝑐 ∨ ∃𝑏 ∈ 𝐵 𝑏 ≤s 𝑌)))

Distinct variable groups:   𝐴,𝑏,𝑐   𝐵,𝑏,𝑐   𝐶,𝑏,𝑐   𝐷,𝑏,𝑐   𝑋,𝑏,𝑐   𝑌,𝑏,𝑐

Proof of Theorem sltrec

Step	Hyp	Ref	Expression
1		simplr 768	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝐶 <<s 𝐷)
2		simpll 766	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝐴 <<s 𝐵)
3		simprr 772	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝑌 = (𝐶 |s 𝐷))
4		simprl 770	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝑋 = (𝐴 |s 𝐵))
5		slerec 27882	. . . . 5 ⊢ (((𝐶 <<s 𝐷 ∧ 𝐴 <<s 𝐵) ∧ (𝑌 = (𝐶 |s 𝐷) ∧ 𝑋 = (𝐴 |s 𝐵))) → (𝑌 ≤s 𝑋 ↔ (∀𝑏 ∈ 𝐵 𝑌 <s 𝑏 ∧ ∀𝑐 ∈ 𝐶 𝑐 <s 𝑋)))
6	1, 2, 3, 4, 5	syl22anc 838	. . . 4 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑌 ≤s 𝑋 ↔ (∀𝑏 ∈ 𝐵 𝑌 <s 𝑏 ∧ ∀𝑐 ∈ 𝐶 𝑐 <s 𝑋)))
7		ancom 460	. . . 4 ⊢ ((∀𝑏 ∈ 𝐵 𝑌 <s 𝑏 ∧ ∀𝑐 ∈ 𝐶 𝑐 <s 𝑋) ↔ (∀𝑐 ∈ 𝐶 𝑐 <s 𝑋 ∧ ∀𝑏 ∈ 𝐵 𝑌 <s 𝑏))
8	6, 7	bitrdi 287	. . 3 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑌 ≤s 𝑋 ↔ (∀𝑐 ∈ 𝐶 𝑐 <s 𝑋 ∧ ∀𝑏 ∈ 𝐵 𝑌 <s 𝑏)))
9		scutcut 27864	. . . . . . 7 ⊢ (𝐶 <<s 𝐷 → ((𝐶 |s 𝐷) ∈ No ∧ 𝐶 <<s {(𝐶 |s 𝐷)} ∧ {(𝐶 |s 𝐷)} <<s 𝐷))
10	9	simp1d 1142	. . . . . 6 ⊢ (𝐶 <<s 𝐷 → (𝐶 |s 𝐷) ∈ No )
11	10	ad2antlr 726	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝐶 |s 𝐷) ∈ No )
12	3, 11	eqeltrd 2844	. . . 4 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝑌 ∈ No )
13		scutcut 27864	. . . . . . 7 ⊢ (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
14	13	simp1d 1142	. . . . . 6 ⊢ (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) ∈ No )
15	14	ad2antrr 725	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝐴 |s 𝐵) ∈ No )
16	4, 15	eqeltrd 2844	. . . 4 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝑋 ∈ No )
17		slenlt 27815	. . . 4 ⊢ ((𝑌 ∈ No ∧ 𝑋 ∈ No ) → (𝑌 ≤s 𝑋 ↔ ¬ 𝑋 <s 𝑌))
18	12, 16, 17	syl2anc 583	. . 3 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑌 ≤s 𝑋 ↔ ¬ 𝑋 <s 𝑌))
19		ssltss1 27851	. . . . . . . . 9 ⊢ (𝐶 <<s 𝐷 → 𝐶 ⊆ No )
20	19	ad2antlr 726	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝐶 ⊆ No )
21	20	sselda 4008	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑐 ∈ 𝐶) → 𝑐 ∈ No )
22	16	adantr 480	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑐 ∈ 𝐶) → 𝑋 ∈ No )
23		sltnle 27816	. . . . . . 7 ⊢ ((𝑐 ∈ No ∧ 𝑋 ∈ No ) → (𝑐 <s 𝑋 ↔ ¬ 𝑋 ≤s 𝑐))
24	21, 22, 23	syl2anc 583	. . . . . 6 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑐 ∈ 𝐶) → (𝑐 <s 𝑋 ↔ ¬ 𝑋 ≤s 𝑐))
25	24	ralbidva 3182	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (∀𝑐 ∈ 𝐶 𝑐 <s 𝑋 ↔ ∀𝑐 ∈ 𝐶 ¬ 𝑋 ≤s 𝑐))
26	12	adantr 480	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑏 ∈ 𝐵) → 𝑌 ∈ No )
27		ssltss2 27852	. . . . . . . . 9 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
28	27	ad2antrr 725	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝐵 ⊆ No )
29	28	sselda 4008	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ No )
30		sltnle 27816	. . . . . . 7 ⊢ ((𝑌 ∈ No ∧ 𝑏 ∈ No ) → (𝑌 <s 𝑏 ↔ ¬ 𝑏 ≤s 𝑌))
31	26, 29, 30	syl2anc 583	. . . . . 6 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑏 ∈ 𝐵) → (𝑌 <s 𝑏 ↔ ¬ 𝑏 ≤s 𝑌))
32	31	ralbidva 3182	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (∀𝑏 ∈ 𝐵 𝑌 <s 𝑏 ↔ ∀𝑏 ∈ 𝐵 ¬ 𝑏 ≤s 𝑌))
33	25, 32	anbi12d 631	. . . 4 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → ((∀𝑐 ∈ 𝐶 𝑐 <s 𝑋 ∧ ∀𝑏 ∈ 𝐵 𝑌 <s 𝑏) ↔ (∀𝑐 ∈ 𝐶 ¬ 𝑋 ≤s 𝑐 ∧ ∀𝑏 ∈ 𝐵 ¬ 𝑏 ≤s 𝑌)))
34		ralnex 3078	. . . . . 6 ⊢ (∀𝑐 ∈ 𝐶 ¬ 𝑋 ≤s 𝑐 ↔ ¬ ∃𝑐 ∈ 𝐶 𝑋 ≤s 𝑐)
35		ralnex 3078	. . . . . 6 ⊢ (∀𝑏 ∈ 𝐵 ¬ 𝑏 ≤s 𝑌 ↔ ¬ ∃𝑏 ∈ 𝐵 𝑏 ≤s 𝑌)
36	34, 35	anbi12i 627	. . . . 5 ⊢ ((∀𝑐 ∈ 𝐶 ¬ 𝑋 ≤s 𝑐 ∧ ∀𝑏 ∈ 𝐵 ¬ 𝑏 ≤s 𝑌) ↔ (¬ ∃𝑐 ∈ 𝐶 𝑋 ≤s 𝑐 ∧ ¬ ∃𝑏 ∈ 𝐵 𝑏 ≤s 𝑌))
37		ioran 984	. . . . 5 ⊢ (¬ (∃𝑐 ∈ 𝐶 𝑋 ≤s 𝑐 ∨ ∃𝑏 ∈ 𝐵 𝑏 ≤s 𝑌) ↔ (¬ ∃𝑐 ∈ 𝐶 𝑋 ≤s 𝑐 ∧ ¬ ∃𝑏 ∈ 𝐵 𝑏 ≤s 𝑌))
38	36, 37	bitr4i 278	. . . 4 ⊢ ((∀𝑐 ∈ 𝐶 ¬ 𝑋 ≤s 𝑐 ∧ ∀𝑏 ∈ 𝐵 ¬ 𝑏 ≤s 𝑌) ↔ ¬ (∃𝑐 ∈ 𝐶 𝑋 ≤s 𝑐 ∨ ∃𝑏 ∈ 𝐵 𝑏 ≤s 𝑌))
39	33, 38	bitrdi 287	. . 3 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → ((∀𝑐 ∈ 𝐶 𝑐 <s 𝑋 ∧ ∀𝑏 ∈ 𝐵 𝑌 <s 𝑏) ↔ ¬ (∃𝑐 ∈ 𝐶 𝑋 ≤s 𝑐 ∨ ∃𝑏 ∈ 𝐵 𝑏 ≤s 𝑌)))
40	8, 18, 39	3bitr3d 309	. 2 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (¬ 𝑋 <s 𝑌 ↔ ¬ (∃𝑐 ∈ 𝐶 𝑋 ≤s 𝑐 ∨ ∃𝑏 ∈ 𝐵 𝑏 ≤s 𝑌)))
41	40	con4bid 317	1 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑋 <s 𝑌 ↔ (∃𝑐 ∈ 𝐶 𝑋 ≤s 𝑐 ∨ ∃𝑏 ∈ 𝐵 𝑏 ≤s 𝑌)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   ⊆ wss 3976  {csn 4648   class class class wbr 5166  (class class class)co 7448   No csur 27702   <s cslt 27703   ≤s csle 27807   <<s csslt 27843   |s cscut 27845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1o 8522  df-2o 8523  df-no 27705  df-slt 27706  df-bday 27707  df-sle 27808  df-sslt 27844  df-scut 27846

This theorem is referenced by:  0slt1s  27892  sltn0  27961  sleadd1  28040  nohalf  28425  halfcut  28434