Theorem scutbdaybnd2lim 34011

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
scutbdaybnd2lim	⊢ ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday ‘(𝐴 |s 𝐵)) ⊆ ∪ ( bday “ (𝐴 ∪ 𝐵)))

Proof of Theorem scutbdaybnd2lim

Step	Hyp	Ref	Expression
1		scutbdaybnd2 34010	. . . 4 ⊢ (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))
2	1	adantr 481	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday ‘(𝐴 |s 𝐵)) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))
3		bdayfun 33967	. . . . . . . . 9 ⊢ Fun bday
4		ssltex1 33981	. . . . . . . . . 10 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
5		ssltex2 33982	. . . . . . . . . 10 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
6		unexg 7599	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
7	4, 5, 6	syl2anc 584	. . . . . . . . 9 ⊢ (𝐴 <<s 𝐵 → (𝐴 ∪ 𝐵) ∈ V)
8		funimaexg 6520	. . . . . . . . 9 ⊢ ((Fun bday ∧ (𝐴 ∪ 𝐵) ∈ V) → ( bday “ (𝐴 ∪ 𝐵)) ∈ V)
9	3, 7, 8	sylancr 587	. . . . . . . 8 ⊢ (𝐴 <<s 𝐵 → ( bday “ (𝐴 ∪ 𝐵)) ∈ V)
10	9	uniexd 7595	. . . . . . 7 ⊢ (𝐴 <<s 𝐵 → ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ V)
11	10	adantr 481	. . . . . 6 ⊢ ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ V)
12		nlimsucg 7689	. . . . . 6 ⊢ (∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ V → ¬ Lim suc ∪ ( bday “ (𝐴 ∪ 𝐵)))
13	11, 12	syl 17	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ¬ Lim suc ∪ ( bday “ (𝐴 ∪ 𝐵)))
14		limeq 6278	. . . . . . 7 ⊢ (( bday ‘(𝐴 |s 𝐵)) = suc ∪ ( bday “ (𝐴 ∪ 𝐵)) → (Lim ( bday ‘(𝐴 |s 𝐵)) ↔ Lim suc ∪ ( bday “ (𝐴 ∪ 𝐵))))
15	14	biimpcd 248	. . . . . 6 ⊢ (Lim ( bday ‘(𝐴 |s 𝐵)) → (( bday ‘(𝐴 |s 𝐵)) = suc ∪ ( bday “ (𝐴 ∪ 𝐵)) → Lim suc ∪ ( bday “ (𝐴 ∪ 𝐵))))
16	15	adantl 482	. . . . 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 197	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ¬ ( bday ‘(𝐴 |s 𝐵)) = suc ∪ ( bday “ (𝐴 ∪ 𝐵)))
18	17	neqned 2950	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday ‘(𝐴 |s 𝐵)) ≠ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))
19		bdayelon 33971	. . . . 5 ⊢ ( bday ‘(𝐴 |s 𝐵)) ∈ On
20	19	onordi 6371	. . . 4 ⊢ Ord ( bday ‘(𝐴 |s 𝐵))
21		imassrn 5980	. . . . . . 7 ⊢ ( bday “ (𝐴 ∪ 𝐵)) ⊆ ran bday
22		bdayrn 33970	. . . . . . 7 ⊢ ran bday = On
23	21, 22	sseqtri 3957	. . . . . 6 ⊢ ( bday “ (𝐴 ∪ 𝐵)) ⊆ On
24		ssorduni 7629	. . . . . 6 ⊢ (( bday “ (𝐴 ∪ 𝐵)) ⊆ On → Ord ∪ ( bday “ (𝐴 ∪ 𝐵)))
25	23, 24	ax-mp 5	. . . . 5 ⊢ Ord ∪ ( bday “ (𝐴 ∪ 𝐵))
26		ordsuc 7661	. . . . 5 ⊢ (Ord ∪ ( bday “ (𝐴 ∪ 𝐵)) ↔ Ord suc ∪ ( bday “ (𝐴 ∪ 𝐵)))
27	25, 26	mpbi 229	. . . 4 ⊢ Ord suc ∪ ( bday “ (𝐴 ∪ 𝐵))
28		ordelssne 6293	. . . 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 689	. . 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 583	. 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 6352	. . 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 586	. 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 256	1 ⊢ ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday ‘(𝐴 |s 𝐵)) ⊆ ∪ ( bday “ (𝐴 ∪ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  Vcvv 3432   ∪ cun 3885   ⊆ wss 3887  ∪ cuni 4839   class class class wbr 5074  ran crn 5590   “ cima 5592  Ord word 6265  Oncon0 6266  Lim wlim 6267  suc csuc 6268  Fun wfun 6427  ‘cfv 6433  (class class class)co 7275   bday cbday 33845   <<s csslt 33975   |s cscut 33977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1o 8297  df-2o 8298  df-no 33846  df-slt 33847  df-bday 33848  df-sslt 33976  df-scut 33978

This theorem is referenced by: (None)