Theorem onnolt 28280

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 775	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) ∧ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → 𝐵 ∈ No )
2		onno 28269	. . . . . . . . 9 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )
3	2	adantr 482	. . . . . . . 8 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → 𝐴 ∈ No )
4	3	adantr 482	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) ∧ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → 𝐴 ∈ No )
5		bdayon 27766	. . . . . . . . . . 11 ⊢ ( bday ‘𝐴) ∈ On
6		simpr 486	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → 𝐵 ∈ No )
7		oldbday 27915	. . . . . . . . . . 11 ⊢ ((( bday ‘𝐴) ∈ On ∧ 𝐵 ∈ No ) → (𝐵 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝐵) ∈ ( bday ‘𝐴)))
8	5, 6, 7	sylancr 594	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → (𝐵 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝐵) ∈ ( bday ‘𝐴)))
9		onleft 28274	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Ons → ( O ‘( bday ‘𝐴)) = ( L ‘𝐴))
10	9	eleq2d 2827	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Ons → (𝐵 ∈ ( O ‘( bday ‘𝐴)) ↔ 𝐵 ∈ ( L ‘𝐴)))
11	10	adantr 482	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → (𝐵 ∈ ( O ‘( bday ‘𝐴)) ↔ 𝐵 ∈ ( L ‘𝐴)))
12	8, 11	bitr3d 283	. . . . . . . . 9 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → (( bday ‘𝐵) ∈ ( bday ‘𝐴) ↔ 𝐵 ∈ ( L ‘𝐴)))
13		leftlt 27867	. . . . . . . . 9 ⊢ (𝐵 ∈ ( L ‘𝐴) → 𝐵 <s 𝐴)
14	12, 13	biimtrdi 255	. . . . . . . 8 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → (( bday ‘𝐵) ∈ ( bday ‘𝐴) → 𝐵 <s 𝐴))
15	14	imp 408	. . . . . . 7 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) ∧ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → 𝐵 <s 𝐴)
16	1, 4, 15	ltlesd 27759	. . . . . 6 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) ∧ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → 𝐵 ≤s 𝐴)
17	16	ex 414	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → (( bday ‘𝐵) ∈ ( bday ‘𝐴) → 𝐵 ≤s 𝐴))
18		leftssold 27885	. . . . . . . 8 ⊢ ( L ‘𝐵) ⊆ ( O ‘( bday ‘𝐵))
19		fveq2 6831	. . . . . . . . . 10 ⊢ (( bday ‘𝐵) = ( bday ‘𝐴) → ( O ‘( bday ‘𝐵)) = ( O ‘( bday ‘𝐴)))
20	19	3ad2ant3 1142	. . . . . . . . 9 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → ( O ‘( bday ‘𝐵)) = ( O ‘( bday ‘𝐴)))
21	9	3ad2ant1 1140	. . . . . . . . 9 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → ( O ‘( bday ‘𝐴)) = ( L ‘𝐴))
22	20, 21	eqtrd 2776	. . . . . . . 8 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → ( O ‘( bday ‘𝐵)) = ( L ‘𝐴))
23	18, 22	sseqtrid 3959	. . . . . . 7 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → ( L ‘𝐵) ⊆ ( L ‘𝐴))
24		simp2 1144	. . . . . . . 8 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → 𝐵 ∈ No )
25	2	3ad2ant1 1140	. . . . . . . 8 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → 𝐴 ∈ No )
26		simp3 1145	. . . . . . . 8 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → ( bday ‘𝐵) = ( bday ‘𝐴))
27		leslss 27923	. . . . . . . 8 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → (𝐵 ≤s 𝐴 ↔ ( L ‘𝐵) ⊆ ( L ‘𝐴)))
28	24, 25, 26, 27	syl3anc 1380	. . . . . . 7 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → (𝐵 ≤s 𝐴 ↔ ( L ‘𝐵) ⊆ ( L ‘𝐴)))
29	23, 28	mpbird 259	. . . . . 6 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → 𝐵 ≤s 𝐴)
30	29	3expia 1128	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → (( bday ‘𝐵) = ( bday ‘𝐴) → 𝐵 ≤s 𝐴))
31	17, 30	jaod 866	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → ((( bday ‘𝐵) ∈ ( bday ‘𝐴) ∨ ( bday ‘𝐵) = ( bday ‘𝐴)) → 𝐵 ≤s 𝐴))
32		bdayon 27766	. . . . . . 7 ⊢ ( bday ‘𝐵) ∈ On
33	32, 5	onsseli 6436	. . . . . 6 ⊢ (( bday ‘𝐵) ⊆ ( bday ‘𝐴) ↔ (( bday ‘𝐵) ∈ ( bday ‘𝐴) ∨ ( bday ‘𝐵) = ( bday ‘𝐴)))
34		ontri1 6348	. . . . . . 7 ⊢ ((( bday ‘𝐵) ∈ On ∧ ( bday ‘𝐴) ∈ On) → (( bday ‘𝐵) ⊆ ( bday ‘𝐴) ↔ ¬ ( bday ‘𝐴) ∈ ( bday ‘𝐵)))
35	32, 5, 34	mp2an 699	. . . . . 6 ⊢ (( bday ‘𝐵) ⊆ ( bday ‘𝐴) ↔ ¬ ( bday ‘𝐴) ∈ ( bday ‘𝐵))
36	33, 35	bitr3i 279	. . . . 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 27738	. . . . 5 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ) → (𝐵 ≤s 𝐴 ↔ ¬ 𝐴 <s 𝐵))
39	6, 3, 38	syl2anc 591	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → (𝐵 ≤s 𝐴 ↔ ¬ 𝐴 <s 𝐵))
40	31, 37, 39	3imtr3d 295	. . 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 1124	1 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ 𝐴 <s 𝐵) → ( bday ‘𝐴) ∈ ( bday ‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 397   ∨ wo 854   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121   ⊆ wss 3885   class class class wbr 5075  Oncon0 6314  ‘cfv 6489   No csur 27625   <s clts 27626   bday cbday 27627   ≤s cles 27730   O cold 27837   L cleft 27839  On_scons 28265

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-1o 8399  df-2o 8400  df-no 27628  df-lts 27629  df-bday 27630  df-les 27731  df-slts 27772  df-cuts 27774  df-made 27841  df-old 27842  df-left 27844  df-right 27845  df-ons 28266

This theorem is referenced by:  onlts  28281  bdayn0p1  28383