Theorem onnolt 28219

Description: If a surreal ordinal is less than a given surreal, then it is simpler. (Contributed by Scott Fenton, 7-Nov-2025.)

Assertion

Ref	Expression
onnolt	⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ 𝐴 <s 𝐵) → ( bday ‘𝐴) ∈ ( bday ‘𝐵))

Proof of Theorem onnolt

Step	Hyp	Ref	Expression
1		simplr 768	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) ∧ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → 𝐵 ∈ No )
2		onsno 28208	. . . . . . . . 9 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )
3	2	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → 𝐴 ∈ No )
4	3	adantr 480	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) ∧ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → 𝐴 ∈ No )
5		bdayelon 27740	. . . . . . . . . . 11 ⊢ ( bday ‘𝐴) ∈ On
6		simpr 484	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → 𝐵 ∈ No )
7		oldbday 27864	. . . . . . . . . . 11 ⊢ ((( bday ‘𝐴) ∈ On ∧ 𝐵 ∈ No ) → (𝐵 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝐵) ∈ ( bday ‘𝐴)))
8	5, 6, 7	sylancr 587	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → (𝐵 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝐵) ∈ ( bday ‘𝐴)))
9		onsleft 28213	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Ons → ( O ‘( bday ‘𝐴)) = ( L ‘𝐴))
10	9	eleq2d 2820	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Ons → (𝐵 ∈ ( O ‘( bday ‘𝐴)) ↔ 𝐵 ∈ ( L ‘𝐴)))
11	10	adantr 480	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → (𝐵 ∈ ( O ‘( bday ‘𝐴)) ↔ 𝐵 ∈ ( L ‘𝐴)))
12	8, 11	bitr3d 281	. . . . . . . . 9 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → (( bday ‘𝐵) ∈ ( bday ‘𝐴) ↔ 𝐵 ∈ ( L ‘𝐴)))
13		leftlt 27827	. . . . . . . . 9 ⊢ (𝐵 ∈ ( L ‘𝐴) → 𝐵 <s 𝐴)
14	12, 13	biimtrdi 253	. . . . . . . 8 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → (( bday ‘𝐵) ∈ ( bday ‘𝐴) → 𝐵 <s 𝐴))
15	14	imp 406	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) ∧ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → 𝐵 <s 𝐴)
16	1, 4, 15	sltled 27733	. . . . . 6 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) ∧ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → 𝐵 ≤s 𝐴)
17	16	ex 412	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → (( bday ‘𝐵) ∈ ( bday ‘𝐴) → 𝐵 ≤s 𝐴))
18		leftssold 27842	. . . . . . . 8 ⊢ ( L ‘𝐵) ⊆ ( O ‘( bday ‘𝐵))
19		fveq2 6876	. . . . . . . . . 10 ⊢ (( bday ‘𝐵) = ( bday ‘𝐴) → ( O ‘( bday ‘𝐵)) = ( O ‘( bday ‘𝐴)))
20	19	3ad2ant3 1135	. . . . . . . . 9 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → ( O ‘( bday ‘𝐵)) = ( O ‘( bday ‘𝐴)))
21	9	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → ( O ‘( bday ‘𝐴)) = ( L ‘𝐴))
22	20, 21	eqtrd 2770	. . . . . . . 8 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → ( O ‘( bday ‘𝐵)) = ( L ‘𝐴))
23	18, 22	sseqtrid 4001	. . . . . . 7 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → ( L ‘𝐵) ⊆ ( L ‘𝐴))
24		simp2 1137	. . . . . . . 8 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → 𝐵 ∈ No )
25	2	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → 𝐴 ∈ No )
26		simp3 1138	. . . . . . . 8 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → ( bday ‘𝐵) = ( bday ‘𝐴))
27		slelss 27872	. . . . . . . 8 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → (𝐵 ≤s 𝐴 ↔ ( L ‘𝐵) ⊆ ( L ‘𝐴)))
28	24, 25, 26, 27	syl3anc 1373	. . . . . . 7 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → (𝐵 ≤s 𝐴 ↔ ( L ‘𝐵) ⊆ ( L ‘𝐴)))
29	23, 28	mpbird 257	. . . . . 6 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → 𝐵 ≤s 𝐴)
30	29	3expia 1121	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → (( bday ‘𝐵) = ( bday ‘𝐴) → 𝐵 ≤s 𝐴))
31	17, 30	jaod 859	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → ((( bday ‘𝐵) ∈ ( bday ‘𝐴) ∨ ( bday ‘𝐵) = ( bday ‘𝐴)) → 𝐵 ≤s 𝐴))
32		bdayelon 27740	. . . . . . 7 ⊢ ( bday ‘𝐵) ∈ On
33	32, 5	onsseli 6475	. . . . . 6 ⊢ (( bday ‘𝐵) ⊆ ( bday ‘𝐴) ↔ (( bday ‘𝐵) ∈ ( bday ‘𝐴) ∨ ( bday ‘𝐵) = ( bday ‘𝐴)))
34		ontri1 6386	. . . . . . 7 ⊢ ((( bday ‘𝐵) ∈ On ∧ ( bday ‘𝐴) ∈ On) → (( bday ‘𝐵) ⊆ ( bday ‘𝐴) ↔ ¬ ( bday ‘𝐴) ∈ ( bday ‘𝐵)))
35	32, 5, 34	mp2an 692	. . . . . 6 ⊢ (( bday ‘𝐵) ⊆ ( bday ‘𝐴) ↔ ¬ ( bday ‘𝐴) ∈ ( bday ‘𝐵))
36	33, 35	bitr3i 277	. . . . 5 ⊢ ((( bday ‘𝐵) ∈ ( bday ‘𝐴) ∨ ( bday ‘𝐵) = ( bday ‘𝐴)) ↔ ¬ ( bday ‘𝐴) ∈ ( bday ‘𝐵))
37	36	a1i 11	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → ((( bday ‘𝐵) ∈ ( bday ‘𝐴) ∨ ( bday ‘𝐵) = ( bday ‘𝐴)) ↔ ¬ ( bday ‘𝐴) ∈ ( bday ‘𝐵)))
38		slenlt 27716	. . . . 5 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ) → (𝐵 ≤s 𝐴 ↔ ¬ 𝐴 <s 𝐵))
39	6, 3, 38	syl2anc 584	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → (𝐵 ≤s 𝐴 ↔ ¬ 𝐴 <s 𝐵))
40	31, 37, 39	3imtr3d 293	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → (¬ ( bday ‘𝐴) ∈ ( bday ‘𝐵) → ¬ 𝐴 <s 𝐵))
41	40	con4d 115	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 → ( bday ‘𝐴) ∈ ( bday ‘𝐵)))
42	41	3impia 1117	1 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ 𝐴 <s 𝐵) → ( bday ‘𝐴) ∈ ( bday ‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108   ⊆ wss 3926   class class class wbr 5119  Oncon0 6352  ‘cfv 6531   No csur 27603   <s cslt 27604   bday cbday 27605   ≤s csle 27708   O cold 27803   L cleft 27805  On_scons 28204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-1o 8480  df-2o 8481  df-no 27606  df-slt 27607  df-bday 27608  df-sle 27709  df-sslt 27745  df-scut 27747  df-made 27807  df-old 27808  df-left 27810  df-right 27811  df-ons 28205

This theorem is referenced by:  onslt  28220  bdayn0p1  28310