Theorem cutbdaybnd2lim 28188

Description: An upper bound on the birthday of a surreal cut when it is a limit birthday. (Contributed by Scott Fenton, 7-Aug-2024.)

Assertion

Ref	Expression
cutbdaybnd2lim	⊢ ((𝐴 <<s 𝐵 ∧ Lim (bday‘(𝐴 |s 𝐵))) → (bday‘(𝐴 |s 𝐵)) ⊆ ∪ (bday “ (𝐴 ∪ 𝐵)))

Proof of Theorem cutbdaybnd2lim

Step	Hyp	Ref	Expression
1		cutbdaybnd2 28187	. . . 4 ⊢ (𝐴 <<s 𝐵 → (bday‘(𝐴 |s 𝐵)) ⊆ suc ∪ (bday “ (𝐴 ∪ 𝐵)))
2	1	adantr 486	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ Lim (bday‘(𝐴 |s 𝐵))) → (bday‘(𝐴 |s 𝐵)) ⊆ suc ∪ (bday “ (𝐴 ∪ 𝐵)))
3		bdayfun 28138	. . . . . . . . 9 ⊢ Fun bday
4		sltsex1 28154	. . . . . . . . . 10 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
5		sltsex2 28155	. . . . . . . . . 10 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
6		unexg 7760	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
7	4, 5, 6	syl2anc 596	. . . . . . . . 9 ⊢ (𝐴 <<s 𝐵 → (𝐴 ∪ 𝐵) ∈ V)
8		funimaexg 6626	. . . . . . . . 9 ⊢ ((Fun bday ∧ (𝐴 ∪ 𝐵) ∈ V) → (bday “ (𝐴 ∪ 𝐵)) ∈ V)
9	3, 7, 8	sylancr 599	. . . . . . . 8 ⊢ (𝐴 <<s 𝐵 → (bday “ (𝐴 ∪ 𝐵)) ∈ V)
10	9	uniexd 7759	. . . . . . 7 ⊢ (𝐴 <<s 𝐵 → ∪ (bday “ (𝐴 ∪ 𝐵)) ∈ V)
11	10	adantr 486	. . . . . 6 ⊢ ((𝐴 <<s 𝐵 ∧ Lim (bday‘(𝐴 |s 𝐵))) → ∪ (bday “ (𝐴 ∪ 𝐵)) ∈ V)
12		nlimsucg 7853	. . . . . 6 ⊢ (∪ (bday “ (𝐴 ∪ 𝐵)) ∈ V → ¬ Lim suc ∪ (bday “ (𝐴 ∪ 𝐵)))
13	11, 12	syl 18	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ Lim (bday‘(𝐴 |s 𝐵))) → ¬ Lim suc ∪ (bday “ (𝐴 ∪ 𝐵)))
14		limeq 6374	. . . . . . 7 ⊢ ((bday‘(𝐴 |s 𝐵)) = suc ∪ (bday “ (𝐴 ∪ 𝐵)) → (Lim (bday‘(𝐴 |s 𝐵)) ↔ Lim suc ∪ (bday “ (𝐴 ∪ 𝐵))))
15	14	biimpcd 252	. . . . . 6 ⊢ (Lim (bday‘(𝐴 |s 𝐵)) → ((bday‘(𝐴 |s 𝐵)) = suc ∪ (bday “ (𝐴 ∪ 𝐵)) → Lim suc ∪ (bday “ (𝐴 ∪ 𝐵))))
16	15	adantl 487	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ Lim (bday‘(𝐴 |s 𝐵))) → ((bday‘(𝐴 |s 𝐵)) = suc ∪ (bday “ (𝐴 ∪ 𝐵)) → Lim suc ∪ (bday “ (𝐴 ∪ 𝐵))))
17	13, 16	mtod 201	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ Lim (bday‘(𝐴 |s 𝐵))) → ¬ (bday‘(𝐴 |s 𝐵)) = suc ∪ (bday “ (𝐴 ∪ 𝐵)))
18	17	neqned 2962	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ Lim (bday‘(𝐴 |s 𝐵))) → (bday‘(𝐴 |s 𝐵)) ≠ suc ∪ (bday “ (𝐴 ∪ 𝐵)))
19		bdayon 28143	. . . . 5 ⊢ (bday‘(𝐴 |s 𝐵)) ∈ On
20	19	onordi 6476	. . . 4 ⊢ Ord (bday‘(𝐴 |s 𝐵))
21		imassrn 6197	. . . . . . 7 ⊢ (bday “ (𝐴 ∪ 𝐵)) ⊆ ran bday
22		bdayrn 28142	. . . . . . 7 ⊢ ran bday = On
23	21, 22	sseqtri 3956	. . . . . 6 ⊢ (bday “ (𝐴 ∪ 𝐵)) ⊆ On
24		ssorduni 7793	. . . . . 6 ⊢ ((bday “ (𝐴 ∪ 𝐵)) ⊆ On → Ord ∪ (bday “ (𝐴 ∪ 𝐵)))
25	23, 24	ax-mp 5	. . . . 5 ⊢ Ord ∪ (bday “ (𝐴 ∪ 𝐵))
26		ordsuc 7825	. . . . 5 ⊢ (Ord ∪ (bday “ (𝐴 ∪ 𝐵)) ↔ Ord suc ∪ (bday “ (𝐴 ∪ 𝐵)))
27	25, 26	mpbi 233	. . . 4 ⊢ Ord suc ∪ (bday “ (𝐴 ∪ 𝐵))
28		ordelssne 6389	. . . 4 ⊢ ((Ord (bday‘(𝐴 |s 𝐵)) ∧ Ord suc ∪ (bday “ (𝐴 ∪ 𝐵))) → ((bday‘(𝐴 |s 𝐵)) ∈ suc ∪ (bday “ (𝐴 ∪ 𝐵)) ↔ ((bday‘(𝐴 |s 𝐵)) ⊆ suc ∪ (bday “ (𝐴 ∪ 𝐵)) ∧ (bday‘(𝐴 |s 𝐵)) ≠ suc ∪ (bday “ (𝐴 ∪ 𝐵)))))
29	20, 27, 28	mp2an 705	. . 3 ⊢ ((bday‘(𝐴 |s 𝐵)) ∈ suc ∪ (bday “ (𝐴 ∪ 𝐵)) ↔ ((bday‘(𝐴 |s 𝐵)) ⊆ suc ∪ (bday “ (𝐴 ∪ 𝐵)) ∧ (bday‘(𝐴 |s 𝐵)) ≠ suc ∪ (bday “ (𝐴 ∪ 𝐵))))
30	2, 18, 29	sylanbrc 595	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ Lim (bday‘(𝐴 |s 𝐵))) → (bday‘(𝐴 |s 𝐵)) ∈ suc ∪ (bday “ (𝐴 ∪ 𝐵)))
31	19	a1i 11	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ Lim (bday‘(𝐴 |s 𝐵))) → (bday‘(𝐴 |s 𝐵)) ∈ On)
32		ordsssuc 6454	. . 3 ⊢ (((bday‘(𝐴 |s 𝐵)) ∈ On ∧ Ord ∪ (bday “ (𝐴 ∪ 𝐵))) → ((bday‘(𝐴 |s 𝐵)) ⊆ ∪ (bday “ (𝐴 ∪ 𝐵)) ↔ (bday‘(𝐴 |s 𝐵)) ∈ suc ∪ (bday “ (𝐴 ∪ 𝐵))))
33	31, 25, 32	sylancl 598	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ Lim (bday‘(𝐴 |s 𝐵))) → ((bday‘(𝐴 |s 𝐵)) ⊆ ∪ (bday “ (𝐴 ∪ 𝐵)) ↔ (bday‘(𝐴 |s 𝐵)) ∈ suc ∪ (bday “ (𝐴 ∪ 𝐵))))
34	30, 33	mpbird 260	1 ⊢ ((𝐴 <<s 𝐵 ∧ Lim (bday‘(𝐴 |s 𝐵))) → (bday‘(𝐴 |s 𝐵)) ⊆ ∪ (bday “ (𝐴 ∪ 𝐵)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  Vcvv 3450   ⊆ wss 3895   ∪ cun 4149  ∪ cuni 4866   class class class wbr 5102  ran crn 5651   “ cima 5653  Ord word 6361  Oncon0 6362  Lim wlim 6363  suc csuc 6364  Fun wfun 6532  ‘cfv 6538  (class class class)co 7420  bdaycbday 28004   <<s cslts 28148   |s ccuts 28150

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 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  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-ord 6365  df-on 6366  df-lim 6367  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-1o 8479  df-2o 8480  df-no 28005  df-lts 28006  df-bday 28007  df-slts 28149  df-cuts 28151

This theorem is used by: (None)