Theorem cutbdaybnd2lim 28070

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 28069	. . . 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 28020	. . . . . . . . 9 ⊢ Fun bday
4		sltsex1 28036	. . . . . . . . . 10 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
5		sltsex2 28037	. . . . . . . . . 10 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
6		unexg 7749	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
7	4, 5, 6	syl2anc 596	. . . . . . . . 9 ⊢ (𝐴 <<s 𝐵 → (𝐴 ∪ 𝐵) ∈ V)
8		funimaexg 6623	. . . . . . . . 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 7748	. . . . . . 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 7842	. . . . . 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 6373	. . . . . . 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 2964	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday ‘(𝐴 |s 𝐵)) ≠ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))
19		bdayon 28025	. . . . 5 ⊢ ( bday ‘(𝐴 |s 𝐵)) ∈ On
20	19	onordi 6475	. . . 4 ⊢ Ord ( bday ‘(𝐴 |s 𝐵))
21		imassrn 6071	. . . . . . 7 ⊢ ( bday “ (𝐴 ∪ 𝐵)) ⊆ ran bday
22		bdayrn 28024	. . . . . . 7 ⊢ ran bday = On
23	21, 22	sseqtri 3982	. . . . . 6 ⊢ ( bday “ (𝐴 ∪ 𝐵)) ⊆ On
24		ssorduni 7782	. . . . . 6 ⊢ (( bday “ (𝐴 ∪ 𝐵)) ⊆ On → Ord ∪ ( bday “ (𝐴 ∪ 𝐵)))
25	23, 24	ax-mp 5	. . . . 5 ⊢ Ord ∪ ( bday “ (𝐴 ∪ 𝐵))
26		ordsuc 7814	. . . . 5 ⊢ (Ord ∪ ( bday “ (𝐴 ∪ 𝐵)) ↔ Ord suc ∪ ( bday “ (𝐴 ∪ 𝐵)))
27	25, 26	mpbi 233	. . . 4 ⊢ Ord suc ∪ ( bday “ (𝐴 ∪ 𝐵))
28		ordelssne 6388	. . . 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 6453	. . 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 1570   ∈ wcel 2145   ≠ wne 2957  Vcvv 3453   ∪ cun 3900   ⊆ wss 3902  ∪ cuni 4870   class class class wbr 5107  ran crn 5660   “ cima 5662  Ord word 6360  Oncon0 6361  Lim wlim 6362  suc csuc 6363  Fun wfun 6531  ‘cfv 6537  (class class class)co 7417   bday cbday 27886   <<s cslts 28030   |s ccuts 28032

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-1o 8459  df-2o 8460  df-no 27887  df-lts 27888  df-bday 27889  df-slts 28031  df-cuts 28033

This theorem is used by: (None)