Theorem sltrec 32513

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 759	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝐶 <<s 𝐷)
2		simpll 757	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝐴 <<s 𝐵)
3		simprr 763	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝑌 = (𝐶 |s 𝐷))
4		simprl 761	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝑋 = (𝐴 |s 𝐵))
5		slerec 32512	. . . . 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 829	. . . 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 454	. . . 4 ⊢ ((∀𝑏 ∈ 𝐵 𝑌 <s 𝑏 ∧ ∀𝑐 ∈ 𝐶 𝑐 <s 𝑋) ↔ (∀𝑐 ∈ 𝐶 𝑐 <s 𝑋 ∧ ∀𝑏 ∈ 𝐵 𝑌 <s 𝑏))
8	6, 7	syl6bb 279	. . 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 32501	. . . . . . 7 ⊢ (𝐶 <<s 𝐷 → ((𝐶 |s 𝐷) ∈ No ∧ 𝐶 <<s {(𝐶 |s 𝐷)} ∧ {(𝐶 |s 𝐷)} <<s 𝐷))
10	9	simp1d 1133	. . . . . 6 ⊢ (𝐶 <<s 𝐷 → (𝐶 |s 𝐷) ∈ No )
11	10	ad2antlr 717	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝐶 |s 𝐷) ∈ No )
12	3, 11	eqeltrd 2859	. . . 4 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝑌 ∈ No )
13		scutcut 32501	. . . . . . 7 ⊢ (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
14	13	simp1d 1133	. . . . . 6 ⊢ (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) ∈ No )
15	14	ad2antrr 716	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝐴 |s 𝐵) ∈ No )
16	4, 15	eqeltrd 2859	. . . 4 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝑋 ∈ No )
17		slenlt 32466	. . . 4 ⊢ ((𝑌 ∈ No ∧ 𝑋 ∈ No ) → (𝑌 ≤s 𝑋 ↔ ¬ 𝑋 <s 𝑌))
18	12, 16, 17	syl2anc 579	. . 3 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑌 ≤s 𝑋 ↔ ¬ 𝑋 <s 𝑌))
19		ssltss1 32492	. . . . . . . . 9 ⊢ (𝐶 <<s 𝐷 → 𝐶 ⊆ No )
20	19	ad2antlr 717	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝐶 ⊆ No )
21	20	sselda 3821	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑐 ∈ 𝐶) → 𝑐 ∈ No )
22	16	adantr 474	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑐 ∈ 𝐶) → 𝑋 ∈ No )
23		sltnle 32467	. . . . . . 7 ⊢ ((𝑐 ∈ No ∧ 𝑋 ∈ No ) → (𝑐 <s 𝑋 ↔ ¬ 𝑋 ≤s 𝑐))
24	21, 22, 23	syl2anc 579	. . . . . 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 3167	. . . . 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 474	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑏 ∈ 𝐵) → 𝑌 ∈ No )
27		ssltss2 32493	. . . . . . . . 9 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
28	27	ad2antrr 716	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝐵 ⊆ No )
29	28	sselda 3821	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ No )
30		sltnle 32467	. . . . . . 7 ⊢ ((𝑌 ∈ No ∧ 𝑏 ∈ No ) → (𝑌 <s 𝑏 ↔ ¬ 𝑏 ≤s 𝑌))
31	26, 29, 30	syl2anc 579	. . . . . 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 3167	. . . . 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 624	. . . 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 3174	. . . . . 6 ⊢ (∀𝑐 ∈ 𝐶 ¬ 𝑋 ≤s 𝑐 ↔ ¬ ∃𝑐 ∈ 𝐶 𝑋 ≤s 𝑐)
35		ralnex 3174	. . . . . 6 ⊢ (∀𝑏 ∈ 𝐵 ¬ 𝑏 ≤s 𝑌 ↔ ¬ ∃𝑏 ∈ 𝐵 𝑏 ≤s 𝑌)
36	34, 35	anbi12i 620	. . . . 5 ⊢ ((∀𝑐 ∈ 𝐶 ¬ 𝑋 ≤s 𝑐 ∧ ∀𝑏 ∈ 𝐵 ¬ 𝑏 ≤s 𝑌) ↔ (¬ ∃𝑐 ∈ 𝐶 𝑋 ≤s 𝑐 ∧ ¬ ∃𝑏 ∈ 𝐵 𝑏 ≤s 𝑌))
37		ioran 969	. . . . 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 270	. . . 4 ⊢ ((∀𝑐 ∈ 𝐶 ¬ 𝑋 ≤s 𝑐 ∧ ∀𝑏 ∈ 𝐵 ¬ 𝑏 ≤s 𝑌) ↔ ¬ (∃𝑐 ∈ 𝐶 𝑋 ≤s 𝑐 ∨ ∃𝑏 ∈ 𝐵 𝑏 ≤s 𝑌))
39	33, 38	syl6bb 279	. . 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 301	. 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 309	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 198   ∧ wa 386   ∨ wo 836   = wceq 1601   ∈ wcel 2107  ∀wral 3090  ∃wrex 3091   ⊆ wss 3792  {csn 4398   class class class wbr 4886  (class class class)co 6922   No csur 32382   <s cslt 32383   ≤s csle 32458   <<s csslt 32485   |s cscut 32487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-ord 5979  df-on 5980  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1o 7843  df-2o 7844  df-no 32385  df-slt 32386  df-bday 32387  df-sle 32459  df-sslt 32486  df-scut 32488

This theorem is referenced by: (None)