Theorem sltrec 33278

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 767	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝐶 <<s 𝐷)
2		simpll 765	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝐴 <<s 𝐵)
3		simprr 771	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝑌 = (𝐶 |s 𝐷))
4		simprl 769	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝑋 = (𝐴 |s 𝐵))
5		slerec 33277	. . . . 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 836	. . . 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 463	. . . 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 289	. . 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 33266	. . . . . . 7 ⊢ (𝐶 <<s 𝐷 → ((𝐶 |s 𝐷) ∈ No ∧ 𝐶 <<s {(𝐶 |s 𝐷)} ∧ {(𝐶 |s 𝐷)} <<s 𝐷))
10	9	simp1d 1138	. . . . . 6 ⊢ (𝐶 <<s 𝐷 → (𝐶 |s 𝐷) ∈ No )
11	10	ad2antlr 725	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝐶 |s 𝐷) ∈ No )
12	3, 11	eqeltrd 2913	. . . 4 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝑌 ∈ No )
13		scutcut 33266	. . . . . . 7 ⊢ (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
14	13	simp1d 1138	. . . . . 6 ⊢ (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) ∈ No )
15	14	ad2antrr 724	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝐴 |s 𝐵) ∈ No )
16	4, 15	eqeltrd 2913	. . . 4 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝑋 ∈ No )
17		slenlt 33231	. . . 4 ⊢ ((𝑌 ∈ No ∧ 𝑋 ∈ No ) → (𝑌 ≤s 𝑋 ↔ ¬ 𝑋 <s 𝑌))
18	12, 16, 17	syl2anc 586	. . 3 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑌 ≤s 𝑋 ↔ ¬ 𝑋 <s 𝑌))
19		ssltss1 33257	. . . . . . . . 9 ⊢ (𝐶 <<s 𝐷 → 𝐶 ⊆ No )
20	19	ad2antlr 725	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝐶 ⊆ No )
21	20	sselda 3967	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑐 ∈ 𝐶) → 𝑐 ∈ No )
22	16	adantr 483	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑐 ∈ 𝐶) → 𝑋 ∈ No )
23		sltnle 33232	. . . . . . 7 ⊢ ((𝑐 ∈ No ∧ 𝑋 ∈ No ) → (𝑐 <s 𝑋 ↔ ¬ 𝑋 ≤s 𝑐))
24	21, 22, 23	syl2anc 586	. . . . . 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 3196	. . . . 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 483	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑏 ∈ 𝐵) → 𝑌 ∈ No )
27		ssltss2 33258	. . . . . . . . 9 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
28	27	ad2antrr 724	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝐵 ⊆ No )
29	28	sselda 3967	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ No )
30		sltnle 33232	. . . . . . 7 ⊢ ((𝑌 ∈ No ∧ 𝑏 ∈ No ) → (𝑌 <s 𝑏 ↔ ¬ 𝑏 ≤s 𝑌))
31	26, 29, 30	syl2anc 586	. . . . . 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 3196	. . . . 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 632	. . . 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 3236	. . . . . 6 ⊢ (∀𝑐 ∈ 𝐶 ¬ 𝑋 ≤s 𝑐 ↔ ¬ ∃𝑐 ∈ 𝐶 𝑋 ≤s 𝑐)
35		ralnex 3236	. . . . . 6 ⊢ (∀𝑏 ∈ 𝐵 ¬ 𝑏 ≤s 𝑌 ↔ ¬ ∃𝑏 ∈ 𝐵 𝑏 ≤s 𝑌)
36	34, 35	anbi12i 628	. . . . 5 ⊢ ((∀𝑐 ∈ 𝐶 ¬ 𝑋 ≤s 𝑐 ∧ ∀𝑏 ∈ 𝐵 ¬ 𝑏 ≤s 𝑌) ↔ (¬ ∃𝑐 ∈ 𝐶 𝑋 ≤s 𝑐 ∧ ¬ ∃𝑏 ∈ 𝐵 𝑏 ≤s 𝑌))
37		ioran 980	. . . . 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 280	. . . 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 289	. . 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 311	. 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 319	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 208   ∧ wa 398   ∨ wo 843   = wceq 1537   ∈ wcel 2114  ∀wral 3138  ∃wrex 3139   ⊆ wss 3936  {csn 4567   class class class wbr 5066  (class class class)co 7156   No csur 33147   <s cslt 33148   ≤s csle 33223   <<s csslt 33250   |s cscut 33252

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-ord 6194  df-on 6195  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1o 8102  df-2o 8103  df-no 33150  df-slt 33151  df-bday 33152  df-sle 33224  df-sslt 33251  df-scut 33253

This theorem is referenced by: (None)