Theorem cutbdaybnd2lim 28213

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 28212	. . . 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 28163	. . . . . . . . 9 ⊢ Fun bday
4		sltsex1 28179	. . . . . . . . . 10 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
5		sltsex2 28180	. . . . . . . . . 10 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
6		unexg 7784	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
7	4, 5, 6	syl2anc 596	. . . . . . . . 9 ⊢ (𝐴 <<s 𝐵 → (𝐴 ∪ 𝐵) ∈ V)
8		funimaexg 6648	. . . . . . . . 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 7783	. . . . . . 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 7877	. . . . . 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 6392	. . . . . . 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 2966	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ Lim (bday‘(𝐴 |s 𝐵))) → (bday‘(𝐴 |s 𝐵)) ≠ suc ∪ (bday “ (𝐴 ∪ 𝐵)))
19		bdayon 28168	. . . . 5 ⊢ (bday‘(𝐴 |s 𝐵)) ∈ On
20	19	onordi 6494	. . . 4 ⊢ Ord (bday‘(𝐴 |s 𝐵))
21		imassrn 6210	. . . . . . 7 ⊢ (bday “ (𝐴 ∪ 𝐵)) ⊆ ran bday
22		bdayrn 28167	. . . . . . 7 ⊢ ran bday = On
23	21, 22	sseqtri 3961	. . . . . 6 ⊢ (bday “ (𝐴 ∪ 𝐵)) ⊆ On
24		ssorduni 7817	. . . . . 6 ⊢ ((bday “ (𝐴 ∪ 𝐵)) ⊆ On → Ord ∪ (bday “ (𝐴 ∪ 𝐵)))
25	23, 24	ax-mp 5	. . . . 5 ⊢ Ord ∪ (bday “ (𝐴 ∪ 𝐵))
26		ordsuc 7849	. . . . 5 ⊢ (Ord ∪ (bday “ (𝐴 ∪ 𝐵)) ↔ Ord suc ∪ (bday “ (𝐴 ∪ 𝐵)))
27	25, 26	mpbi 233	. . . 4 ⊢ Ord suc ∪ (bday “ (𝐴 ∪ 𝐵))
28		ordelssne 6407	. . . 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 6472	. . 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 2147   ≠ wne 2959  Vcvv 3454   ⊆ wss 3900   ∪ cun 4152  ∪ cuni 4870   class class class wbr 5106  ran crn 5656   “ cima 5658  Ord word 6379  Oncon0 6380  Lim wlim 6381  suc csuc 6382  Fun wfun 6552  ‘cfv 6558  (class class class)co 7443  bdaycbday 28029   <<s cslts 28173   |s ccuts 28175

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1o 8504  df-2o 8505  df-no 28030  df-lts 28031  df-bday 28032  df-slts 28174  df-cuts 28176

This theorem is used by: (None)