Theorem scutbdaybnd2lim 27736

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 27735	. . . 4 ⊢ (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))
2	1	adantr 480	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday ‘(𝐴 |s 𝐵)) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))
3		bdayfun 27691	. . . . . . . . 9 ⊢ Fun bday
4		ssltex1 27705	. . . . . . . . . 10 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
5		ssltex2 27706	. . . . . . . . . 10 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
6		unexg 7722	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
7	4, 5, 6	syl2anc 584	. . . . . . . . 9 ⊢ (𝐴 <<s 𝐵 → (𝐴 ∪ 𝐵) ∈ V)
8		funimaexg 6606	. . . . . . . . 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 7721	. . . . . . 7 ⊢ (𝐴 <<s 𝐵 → ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ V)
11	10	adantr 480	. . . . . 6 ⊢ ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ V)
12		nlimsucg 7821	. . . . . 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 6347	. . . . . . 7 ⊢ (( bday ‘(𝐴 |s 𝐵)) = suc ∪ ( bday “ (𝐴 ∪ 𝐵)) → (Lim ( bday ‘(𝐴 |s 𝐵)) ↔ Lim suc ∪ ( bday “ (𝐴 ∪ 𝐵))))
15	14	biimpcd 249	. . . . . 6 ⊢ (Lim ( bday ‘(𝐴 |s 𝐵)) → (( bday ‘(𝐴 |s 𝐵)) = suc ∪ ( bday “ (𝐴 ∪ 𝐵)) → Lim suc ∪ ( bday “ (𝐴 ∪ 𝐵))))
16	15	adantl 481	. . . . 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 198	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ¬ ( bday ‘(𝐴 |s 𝐵)) = suc ∪ ( bday “ (𝐴 ∪ 𝐵)))
18	17	neqned 2933	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday ‘(𝐴 |s 𝐵)) ≠ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))
19		bdayelon 27695	. . . . 5 ⊢ ( bday ‘(𝐴 |s 𝐵)) ∈ On
20	19	onordi 6448	. . . 4 ⊢ Ord ( bday ‘(𝐴 |s 𝐵))
21		imassrn 6045	. . . . . . 7 ⊢ ( bday “ (𝐴 ∪ 𝐵)) ⊆ ran bday
22		bdayrn 27694	. . . . . . 7 ⊢ ran bday = On
23	21, 22	sseqtri 3998	. . . . . 6 ⊢ ( bday “ (𝐴 ∪ 𝐵)) ⊆ On
24		ssorduni 7758	. . . . . 6 ⊢ (( bday “ (𝐴 ∪ 𝐵)) ⊆ On → Ord ∪ ( bday “ (𝐴 ∪ 𝐵)))
25	23, 24	ax-mp 5	. . . . 5 ⊢ Ord ∪ ( bday “ (𝐴 ∪ 𝐵))
26		ordsuc 7791	. . . . 5 ⊢ (Ord ∪ ( bday “ (𝐴 ∪ 𝐵)) ↔ Ord suc ∪ ( bday “ (𝐴 ∪ 𝐵)))
27	25, 26	mpbi 230	. . . 4 ⊢ Ord suc ∪ ( bday “ (𝐴 ∪ 𝐵))
28		ordelssne 6362	. . . 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 692	. . 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 6426	. . 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 257	1 ⊢ ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday ‘(𝐴 |s 𝐵)) ⊆ ∪ ( bday “ (𝐴 ∪ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  Vcvv 3450   ∪ cun 3915   ⊆ wss 3917  ∪ cuni 4874   class class class wbr 5110  ran crn 5642   “ cima 5644  Ord word 6334  Oncon0 6335  Lim wlim 6336  suc csuc 6337  Fun wfun 6508  ‘cfv 6514  (class class class)co 7390   bday cbday 27560   <<s csslt 27699   |s cscut 27701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4914  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1o 8437  df-2o 8438  df-no 27561  df-slt 27562  df-bday 27563  df-sslt 27700  df-scut 27702

This theorem is referenced by: (None)