Theorem onnolt 28679

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 28668	. . . . . . . . 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 28165	. . . . . . . . . . 11 ⊢ (bday‘𝐴) ∈ On
6		simpr 490	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No) → 𝐵 ∈ No)
7		oldbday 28314	. . . . . . . . . . 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 28673	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Ons → (O‘(bday‘𝐴)) = (L‘𝐴))
10	9	eleq2d 2850	. . . . . . . . . . 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 28266	. . . . . . . . 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 28157	. . . . . 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 28284	. . . . . . . 8 ⊢ (L‘𝐵) ⊆ (O‘(bday‘𝐵))
19		fveq2 6905	. . . . . . . . . 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 2798	. . . . . . . 8 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ (bday‘𝐵) = (bday‘𝐴)) → (O‘(bday‘𝐵)) = (L‘𝐴))
23	18, 22	sseqtrid 3955	. . . . . . 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 28322	. . . . . . . 8 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ∧ (bday‘𝐵) = (bday‘𝐴)) → (𝐵 ≤s 𝐴 ↔ (L‘𝐵) ⊆ (L‘𝐴)))
28	24, 25, 26, 27	syl3anc 1399	. . . . . . 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 28165	. . . . . . 7 ⊢ (bday‘𝐵) ∈ On
33	32, 5	onsseli 6500	. . . . . 6 ⊢ ((bday‘𝐵) ⊆ (bday‘𝐴) ↔ ((bday‘𝐵) ∈ (bday‘𝐴) ∨ (bday‘𝐵) = (bday‘𝐴)))
34		ontri1 6412	. . . . . . 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 28136	. . . . 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 2147   ⊆ wss 3900   class class class wbr 5106  Oncon0 6377  ‘cfv 6555  Nocsur 28024   <s clts 28025  bdaycbday 28026   ≤s cles 28128  Ocold 28236  Lcleft 28238  On_scons 28664

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-1o 8501  df-2o 8502  df-no 28027  df-lts 28028  df-bday 28029  df-les 28129  df-slts 28171  df-cuts 28173  df-made 28240  df-old 28241  df-left 28243  df-right 28244  df-ons 28665

This theorem is used by:  onlts  28680  bdayn0p1  28782