Theorem onnolt 28652

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 28641	. . . . . . . . 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 28138	. . . . . . . . . . 11 ⊢ (bday‘𝐴) ∈ On
6		simpr 490	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No) → 𝐵 ∈ No)
7		oldbday 28287	. . . . . . . . . . 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 28646	. . . . . . . . . . . 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 28239	. . . . . . . . 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 28130	. . . . . 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 28257	. . . . . . . 8 ⊢ (L‘𝐵) ⊆ (O‘(bday‘𝐵))
19		fveq2 6885	. . . . . . . . . 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 3950	. . . . . . 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 28295	. . . . . . . 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 28138	. . . . . . 7 ⊢ (bday‘𝐵) ∈ On
33	32, 5	onsseli 6485	. . . . . 6 ⊢ ((bday‘𝐵) ⊆ (bday‘𝐴) ↔ ((bday‘𝐵) ∈ (bday‘𝐴) ∨ (bday‘𝐵) = (bday‘𝐴)))
34		ontri1 6397	. . . . . . 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 28109	. . . . 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 1569   ∈ wcel 2144   ⊆ wss 3895   class class class wbr 5102  Oncon0 6362  ‘cfv 6538  Nocsur 27997   <s clts 27998  bdaycbday 27999   ≤s cles 28101  Ocold 28209  Lcleft 28211  On_scons 28637

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-1o 8476  df-2o 8477  df-no 28000  df-lts 28001  df-bday 28002  df-les 28102  df-slts 28144  df-cuts 28146  df-made 28213  df-old 28214  df-left 28216  df-right 28217  df-ons 28638

This theorem is used by:  onlts  28653  bdayn0p1  28755