Theorem onnolt 28276

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 28265	. . . . . . . . 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 27762	. . . . . . . . . . 11 ⊢ ( bday ‘𝐴) ∈ On
6		simpr 484	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ) → 𝐵 ∈ No )
7		oldbday 27911	. . . . . . . . . . 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 28270	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Ons → ( O ‘( bday ‘𝐴)) = ( L ‘𝐴))
10	9	eleq2d 2823	. . . . . . . . . . 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 27863	. . . . . . . . 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 27755	. . . . . 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 27881	. . . . . . . 8 ⊢ ( L ‘𝐵) ⊆ ( O ‘( bday ‘𝐵))
19		fveq2 6836	. . . . . . . . . 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 2772	. . . . . . . 8 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ No ∧ ( bday ‘𝐵) = ( bday ‘𝐴)) → ( O ‘( bday ‘𝐵)) = ( L ‘𝐴))
23	18, 22	sseqtrid 3965	. . . . . . 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 27919	. . . . . . . 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 27762	. . . . . . 7 ⊢ ( bday ‘𝐵) ∈ On
33	32, 5	onsseli 6441	. . . . . 6 ⊢ (( bday ‘𝐵) ⊆ ( bday ‘𝐴) ↔ (( bday ‘𝐵) ∈ ( bday ‘𝐴) ∨ ( bday ‘𝐵) = ( bday ‘𝐴)))
34		ontri1 6353	. . . . . . 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 27734	. . . . 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 3890   class class class wbr 5086  Oncon0 6319  ‘cfv 6494   No csur 27621   <s clts 27622   bday cbday 27623   ≤s cles 27726   O cold 27833   L cleft 27835  On_scons 28261

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 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-1o 8400  df-2o 8401  df-no 27624  df-lts 27625  df-bday 27626  df-les 27727  df-slts 27768  df-cuts 27770  df-made 27837  df-old 27838  df-left 27840  df-right 27841  df-ons 28262

This theorem is referenced by:  onlts  28277  bdayn0p1  28379