Theorem onnolt 28258

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 769	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) ∧ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → 𝐵 ∈ No )
2		onno 28247	. . . . . . . . 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		bdayon 27744	. . . . . . . . . . 11 ⊢ ( bday ‘𝐴) ∈ On
6		simpr 484	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → 𝐵 ∈ No )
7		oldbday 27893	. . . . . . . . . . 11 ⊢ ((( bday ‘𝐴) ∈ On ∧ 𝐵 ∈ No ) → (𝐵 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝐵) ∈ ( bday ‘𝐴)))
8	5, 6, 7	sylancr 588	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → (𝐵 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝐵) ∈ ( bday ‘𝐴)))
9		onleft 28252	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Ons → ( O ‘( bday ‘𝐴)) = ( L ‘𝐴))
10	9	eleq2d 2822	. . . . . . . . . . 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 27845	. . . . . . . . 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	ltlesd 27737	. . . . . 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 27863	. . . . . . . 8 ⊢ ( L ‘𝐵) ⊆ ( O ‘( bday ‘𝐵))
19		fveq2 6840	. . . . . . . . . 10 ⊢ (( bday ‘𝐵) = ( bday ‘𝐴) → ( O ‘( bday ‘𝐵)) = ( O ‘( bday ‘𝐴)))
20	19	3ad2ant3 1136	. . . . . . . . 9 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → ( O ‘( bday ‘𝐵)) = ( O ‘( bday ‘𝐴)))
21	9	3ad2ant1 1134	. . . . . . . . 9 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → ( O ‘( bday ‘𝐴)) = ( L ‘𝐴))
22	20, 21	eqtrd 2771	. . . . . . . 8 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → ( O ‘( bday ‘𝐵)) = ( L ‘𝐴))
23	18, 22	sseqtrid 3964	. . . . . . 7 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → ( L ‘𝐵) ⊆ ( L ‘𝐴))
24		simp2 1138	. . . . . . . 8 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → 𝐵 ∈ No )
25	2	3ad2ant1 1134	. . . . . . . 8 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → 𝐴 ∈ No )
26		simp3 1139	. . . . . . . 8 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → ( bday ‘𝐵) = ( bday ‘𝐴))
27		leslss 27901	. . . . . . . 8 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → (𝐵 ≤s 𝐴 ↔ ( L ‘𝐵) ⊆ ( L ‘𝐴)))
28	24, 25, 26, 27	syl3anc 1374	. . . . . . 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 1122	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → (( bday ‘𝐵) = ( bday ‘𝐴) → 𝐵 ≤s 𝐴))
31	17, 30	jaod 860	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → ((( bday ‘𝐵) ∈ ( bday ‘𝐴) ∨ ( bday ‘𝐵) = ( bday ‘𝐴)) → 𝐵 ≤s 𝐴))
32		bdayon 27744	. . . . . . 7 ⊢ ( bday ‘𝐵) ∈ On
33	32, 5	onsseli 6445	. . . . . 6 ⊢ (( bday ‘𝐵) ⊆ ( bday ‘𝐴) ↔ (( bday ‘𝐵) ∈ ( bday ‘𝐴) ∨ ( bday ‘𝐵) = ( bday ‘𝐴)))
34		ontri1 6357	. . . . . . 7 ⊢ ((( bday ‘𝐵) ∈ On ∧ ( bday ‘𝐴) ∈ On) → (( bday ‘𝐵) ⊆ ( bday ‘𝐴) ↔ ¬ ( bday ‘𝐴) ∈ ( bday ‘𝐵)))
35	32, 5, 34	mp2an 693	. . . . . 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		lenlts 27716	. . . . 5 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ) → (𝐵 ≤s 𝐴 ↔ ¬ 𝐴 <s 𝐵))
39	6, 3, 38	syl2anc 585	. . . 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 1118	1 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ 𝐴 <s 𝐵) → ( bday ‘𝐴) ∈ ( bday ‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ⊆ wss 3889   class class class wbr 5085  Oncon0 6323  ‘cfv 6498   No csur 27603   <s clts 27604   bday cbday 27605   ≤s cles 27708   O cold 27815   L cleft 27817  On_scons 28243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-1o 8405  df-2o 8406  df-no 27606  df-lts 27607  df-bday 27608  df-les 27709  df-slts 27750  df-cuts 27752  df-made 27819  df-old 27820  df-left 27822  df-right 27823  df-ons 28244

This theorem is referenced by:  onlts  28259  bdayn0p1  28361