Theorem onnolt 28571

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 781	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) ∧ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → 𝐵 ∈ No )
2		onno 28560	. . . . . . . . 9 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )
3	2	adantr 486	. . . . . . . 8 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → 𝐴 ∈ No )
4	3	adantr 486	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) ∧ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → 𝐴 ∈ No )
5		bdayon 28057	. . . . . . . . . . 11 ⊢ ( bday ‘𝐴) ∈ On
6		simpr 490	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → 𝐵 ∈ No )
7		oldbday 28206	. . . . . . . . . . 11 ⊢ ((( bday ‘𝐴) ∈ On ∧ 𝐵 ∈ No ) → (𝐵 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝐵) ∈ ( bday ‘𝐴)))
8	5, 6, 7	sylancr 599	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → (𝐵 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝐵) ∈ ( bday ‘𝐴)))
9		onleft 28565	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Ons → ( O ‘( bday ‘𝐴)) = ( L ‘𝐴))
10	9	eleq2d 2846	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Ons → (𝐵 ∈ ( O ‘( bday ‘𝐴)) ↔ 𝐵 ∈ ( L ‘𝐴)))
11	10	adantr 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → (𝐵 ∈ ( O ‘( bday ‘𝐴)) ↔ 𝐵 ∈ ( L ‘𝐴)))
12	8, 11	bitr3d 284	. . . . . . . . 9 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → (( bday ‘𝐵) ∈ ( bday ‘𝐴) ↔ 𝐵 ∈ ( L ‘𝐴)))
13		leftlt 28158	. . . . . . . . 9 ⊢ (𝐵 ∈ ( L ‘𝐴) → 𝐵 <s 𝐴)
14	12, 13	biimtrdi 256	. . . . . . . 8 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → (( bday ‘𝐵) ∈ ( bday ‘𝐴) → 𝐵 <s 𝐴))
15	14	imp 412	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) ∧ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → 𝐵 <s 𝐴)
16	1, 4, 15	ltlesd 28049	. . . . . 6 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) ∧ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → 𝐵 ≤s 𝐴)
17	16	ex 418	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → (( bday ‘𝐵) ∈ ( bday ‘𝐴) → 𝐵 ≤s 𝐴))
18		leftssold 28176	. . . . . . . 8 ⊢ ( L ‘𝐵) ⊆ ( O ‘( bday ‘𝐵))
19		fveq2 6874	. . . . . . . . . 10 ⊢ (( bday ‘𝐵) = ( bday ‘𝐴) → ( O ‘( bday ‘𝐵)) = ( O ‘( bday ‘𝐴)))
20	19	3ad2ant3 1153	. . . . . . . . 9 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → ( O ‘( bday ‘𝐵)) = ( O ‘( bday ‘𝐴)))
21	9	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → ( O ‘( bday ‘𝐴)) = ( L ‘𝐴))
22	20, 21	eqtrd 2795	. . . . . . . 8 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → ( O ‘( bday ‘𝐵)) = ( L ‘𝐴))
23	18, 22	sseqtrid 3973	. . . . . . 7 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → ( L ‘𝐵) ⊆ ( L ‘𝐴))
24		simp2 1155	. . . . . . . 8 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → 𝐵 ∈ No )
25	2	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → 𝐴 ∈ No )
26		simp3 1156	. . . . . . . 8 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → ( bday ‘𝐵) = ( bday ‘𝐴))
27		leslss 28214	. . . . . . . 8 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → (𝐵 ≤s 𝐴 ↔ ( L ‘𝐵) ⊆ ( L ‘𝐴)))
28	24, 25, 26, 27	syl3anc 1398	. . . . . . 7 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → (𝐵 ≤s 𝐴 ↔ ( L ‘𝐵) ⊆ ( L ‘𝐴)))
29	23, 28	mpbird 260	. . . . . 6 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → 𝐵 ≤s 𝐴)
30	29	3expia 1139	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → (( bday ‘𝐵) = ( bday ‘𝐴) → 𝐵 ≤s 𝐴))
31	17, 30	jaod 873	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → ((( bday ‘𝐵) ∈ ( bday ‘𝐴) ∨ ( bday ‘𝐵) = ( bday ‘𝐴)) → 𝐵 ≤s 𝐴))
32		bdayon 28057	. . . . . . 7 ⊢ ( bday ‘𝐵) ∈ On
33	32, 5	onsseli 6475	. . . . . 6 ⊢ (( bday ‘𝐵) ⊆ ( bday ‘𝐴) ↔ (( bday ‘𝐵) ∈ ( bday ‘𝐴) ∨ ( bday ‘𝐵) = ( bday ‘𝐴)))
34		ontri1 6387	. . . . . . 7 ⊢ ((( bday ‘𝐵) ∈ On ∧ ( bday ‘𝐴) ∈ On) → (( bday ‘𝐵) ⊆ ( bday ‘𝐴) ↔ ¬ ( bday ‘𝐴) ∈ ( bday ‘𝐵)))
35	32, 5, 34	mp2an 705	. . . . . 6 ⊢ (( bday ‘𝐵) ⊆ ( bday ‘𝐴) ↔ ¬ ( bday ‘𝐴) ∈ ( bday ‘𝐵))
36	33, 35	bitr3i 280	. . . . 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		lenlts 28028	. . . . 5 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ) → (𝐵 ≤s 𝐴 ↔ ¬ 𝐴 <s 𝐵))
39	6, 3, 38	syl2anc 596	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → (𝐵 ≤s 𝐴 ↔ ¬ 𝐴 <s 𝐵))
40	31, 37, 39	3imtr3d 296	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → (¬ ( bday ‘𝐴) ∈ ( bday ‘𝐵) → ¬ 𝐴 <s 𝐵))
41	40	con4d 116	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 → ( bday ‘𝐴) ∈ ( bday ‘𝐵)))
42	41	3impia 1135	1 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ 𝐴 <s 𝐵) → ( bday ‘𝐴) ∈ ( bday ‘𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ⊆ wss 3899   class class class wbr 5103  Oncon0 6352  ‘cfv 6528   No csur 27916   <s clts 27917   bday cbday 27918   ≤s cles 28020   O cold 28128   L cleft 28130  On_scons 28556

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6294  df-ord 6355  df-on 6356  df-suc 6358  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-riota 7366  df-ov 7412  df-oprab 7413  df-mpo 7414  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-1o 8455  df-2o 8456  df-no 27919  df-lts 27920  df-bday 27921  df-les 28021  df-slts 28063  df-cuts 28065  df-made 28132  df-old 28133  df-left 28135  df-right 28136  df-ons 28557

This theorem is used by:  onlts  28572  bdayn0p1  28674