Theorem scutbdaybnd2lim 27759

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 27758	. . . 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 27712	. . . . . . . . 9 ⊢ Fun bday
4		ssltex1 27727	. . . . . . . . . 10 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
5		ssltex2 27728	. . . . . . . . . 10 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
6		unexg 7682	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
7	4, 5, 6	syl2anc 584	. . . . . . . . 9 ⊢ (𝐴 <<s 𝐵 → (𝐴 ∪ 𝐵) ∈ V)
8		funimaexg 6573	. . . . . . . . 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 7681	. . . . . . 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 7778	. . . . . 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 6323	. . . . . . 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 2936	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday ‘(𝐴 |s 𝐵)) ≠ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))
19		bdayelon 27716	. . . . 5 ⊢ ( bday ‘(𝐴 |s 𝐵)) ∈ On
20	19	onordi 6424	. . . 4 ⊢ Ord ( bday ‘(𝐴 |s 𝐵))
21		imassrn 6024	. . . . . . 7 ⊢ ( bday “ (𝐴 ∪ 𝐵)) ⊆ ran bday
22		bdayrn 27715	. . . . . . 7 ⊢ ran bday = On
23	21, 22	sseqtri 3979	. . . . . 6 ⊢ ( bday “ (𝐴 ∪ 𝐵)) ⊆ On
24		ssorduni 7718	. . . . . 6 ⊢ (( bday “ (𝐴 ∪ 𝐵)) ⊆ On → Ord ∪ ( bday “ (𝐴 ∪ 𝐵)))
25	23, 24	ax-mp 5	. . . . 5 ⊢ Ord ∪ ( bday “ (𝐴 ∪ 𝐵))
26		ordsuc 7750	. . . . 5 ⊢ (Ord ∪ ( bday “ (𝐴 ∪ 𝐵)) ↔ Ord suc ∪ ( bday “ (𝐴 ∪ 𝐵)))
27	25, 26	mpbi 230	. . . 4 ⊢ Ord suc ∪ ( bday “ (𝐴 ∪ 𝐵))
28		ordelssne 6338	. . . 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 6402	. . 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 1541   ∈ wcel 2113   ≠ wne 2929  Vcvv 3437   ∪ cun 3896   ⊆ wss 3898  ∪ cuni 4858   class class class wbr 5093  ran crn 5620   “ cima 5622  Ord word 6310  Oncon0 6311  Lim wlim 6312  suc csuc 6313  Fun wfun 6480  ‘cfv 6486  (class class class)co 7352   bday cbday 27581   <<s csslt 27721   |s cscut 27723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-uni 4859  df-int 4898  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1o 8391  df-2o 8392  df-no 27582  df-slt 27583  df-bday 27584  df-sslt 27722  df-scut 27724

This theorem is referenced by: (None)