Theorem cutbdaybnd2lim 27968

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 27967	. . . 4 ⊢ (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))
2	1	adantr 485	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday ‘(𝐴 |s 𝐵)) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))
3		bdayfun 27918	. . . . . . . . 9 ⊢ Fun bday
4		sltsex1 27934	. . . . . . . . . 10 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
5		sltsex2 27935	. . . . . . . . . 10 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
6		unexg 7743	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
7	4, 5, 6	syl2anc 595	. . . . . . . . 9 ⊢ (𝐴 <<s 𝐵 → (𝐴 ∪ 𝐵) ∈ V)
8		funimaexg 6624	. . . . . . . . 9 ⊢ ((Fun bday ∧ (𝐴 ∪ 𝐵) ∈ V) → ( bday “ (𝐴 ∪ 𝐵)) ∈ V)
9	3, 7, 8	sylancr 598	. . . . . . . 8 ⊢ (𝐴 <<s 𝐵 → ( bday “ (𝐴 ∪ 𝐵)) ∈ V)
10	9	uniexd 7742	. . . . . . 7 ⊢ (𝐴 <<s 𝐵 → ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ V)
11	10	adantr 485	. . . . . 6 ⊢ ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ V)
12		nlimsucg 7839	. . . . . 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 486	. . . . 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 2965	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday ‘(𝐴 |s 𝐵)) ≠ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))
19		bdayon 27923	. . . . 5 ⊢ ( bday ‘(𝐴 |s 𝐵)) ∈ On
20	19	onordi 6476	. . . 4 ⊢ Ord ( bday ‘(𝐴 |s 𝐵))
21		imassrn 6075	. . . . . . 7 ⊢ ( bday “ (𝐴 ∪ 𝐵)) ⊆ ran bday
22		bdayrn 27922	. . . . . . 7 ⊢ ran bday = On
23	21, 22	sseqtri 3986	. . . . . 6 ⊢ ( bday “ (𝐴 ∪ 𝐵)) ⊆ On
24		ssorduni 7779	. . . . . 6 ⊢ (( bday “ (𝐴 ∪ 𝐵)) ⊆ On → Ord ∪ ( bday “ (𝐴 ∪ 𝐵)))
25	23, 24	ax-mp 5	. . . . 5 ⊢ Ord ∪ ( bday “ (𝐴 ∪ 𝐵))
26		ordsuc 7811	. . . . 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 704	. . 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 594	. 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 597	. 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)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  Vcvv 3455   ∪ cun 3904   ⊆ wss 3906  ∪ cuni 4873   class class class wbr 5110  ran crn 5664   “ cima 5666  Ord word 6361  Oncon0 6362  Lim wlim 6363  suc csuc 6364  Fun wfun 6532  ‘cfv 6538  (class class class)co 7412   bday cbday 27784   <<s cslts 27928   |s ccuts 27930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4874  df-int 4914  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  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 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1o 8454  df-2o 8455  df-no 27785  df-lts 27786  df-bday 27787  df-slts 27929  df-cuts 27931

This theorem is referenced by: (None)