Theorem noetalem4 31866

Description: Lemma for noeta 31868. Bound the birthday of 𝑍 above. (Contributed by Scott Fenton, 6-Dec-2021.)

Hypotheses

Ref	Expression
noetalem.1	⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))
noetalem.2	⊢ 𝑍 = (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1𝑜}))

Assertion

Ref	Expression
noetalem4	⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → ( bday ‘𝑍) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))

Distinct variable group:   𝐴,𝑔,𝑢,𝑣,𝑥,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑆(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑍(𝑥,𝑦,𝑣,𝑢,𝑔)

Proof of Theorem noetalem4

Step	Hyp	Ref	Expression
1		noetalem.1	. . . . . . 7 ⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))
2	1	nosupno 31849	. . . . . 6 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → 𝑆 ∈ No )
3		bdayval 31801	. . . . . 6 ⊢ (𝑆 ∈ No → ( bday ‘𝑆) = dom 𝑆)
4	2, 3	syl 17	. . . . 5 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → ( bday ‘𝑆) = dom 𝑆)
5	1	nosupbday 31851	. . . . 5 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → ( bday ‘𝑆) ⊆ suc ∪ ( bday “ 𝐴))
6	4, 5	eqsstr3d 3640	. . . 4 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → dom 𝑆 ⊆ suc ∪ ( bday “ 𝐴))
7	6	adantr 481	. . 3 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → dom 𝑆 ⊆ suc ∪ ( bday “ 𝐴))
8		unss1 3782	. . 3 ⊢ (dom 𝑆 ⊆ suc ∪ ( bday “ 𝐴) → (dom 𝑆 ∪ suc ∪ ( bday “ 𝐵)) ⊆ (suc ∪ ( bday “ 𝐴) ∪ suc ∪ ( bday “ 𝐵)))
9	7, 8	syl 17	. 2 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → (dom 𝑆 ∪ suc ∪ ( bday “ 𝐵)) ⊆ (suc ∪ ( bday “ 𝐴) ∪ suc ∪ ( bday “ 𝐵)))
10		simpll 790	. . . . 5 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → 𝐴 ⊆ No )
11		simplr 792	. . . . 5 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → 𝐴 ∈ V)
12		simprr 796	. . . . 5 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → 𝐵 ∈ V)
13		noetalem.2	. . . . . 6 ⊢ 𝑍 = (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1𝑜}))
14	1, 13	noetalem1 31863	. . . . 5 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝑍 ∈ No )
15	10, 11, 12, 14	syl3anc 1326	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → 𝑍 ∈ No )
16		bdayval 31801	. . . 4 ⊢ (𝑍 ∈ No → ( bday ‘𝑍) = dom 𝑍)
17	15, 16	syl 17	. . 3 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → ( bday ‘𝑍) = dom 𝑍)
18	13	dmeqi 5325	. . . 4 ⊢ dom 𝑍 = dom (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1𝑜}))
19		dmun 5331	. . . . 5 ⊢ dom (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1𝑜})) = (dom 𝑆 ∪ dom ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1𝑜}))
20		1on 7567	. . . . . . . . . 10 ⊢ 1𝑜 ∈ On
21	20	elexi 3213	. . . . . . . . 9 ⊢ 1𝑜 ∈ V
22	21	snnz 4309	. . . . . . . 8 ⊢ {1𝑜} ≠ ∅
23		dmxp 5344	. . . . . . . 8 ⊢ ({1𝑜} ≠ ∅ → dom ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1𝑜}) = (suc ∪ ( bday “ 𝐵) ∖ dom 𝑆))
24	22, 23	ax-mp 5	. . . . . . 7 ⊢ dom ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1𝑜}) = (suc ∪ ( bday “ 𝐵) ∖ dom 𝑆)
25	24	uneq2i 3764	. . . . . 6 ⊢ (dom 𝑆 ∪ dom ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1𝑜})) = (dom 𝑆 ∪ (suc ∪ ( bday “ 𝐵) ∖ dom 𝑆))
26		undif2 4044	. . . . . 6 ⊢ (dom 𝑆 ∪ (suc ∪ ( bday “ 𝐵) ∖ dom 𝑆)) = (dom 𝑆 ∪ suc ∪ ( bday “ 𝐵))
27	25, 26	eqtri 2644	. . . . 5 ⊢ (dom 𝑆 ∪ dom ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1𝑜})) = (dom 𝑆 ∪ suc ∪ ( bday “ 𝐵))
28	19, 27	eqtri 2644	. . . 4 ⊢ dom (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1𝑜})) = (dom 𝑆 ∪ suc ∪ ( bday “ 𝐵))
29	18, 28	eqtri 2644	. . 3 ⊢ dom 𝑍 = (dom 𝑆 ∪ suc ∪ ( bday “ 𝐵))
30	17, 29	syl6eq 2672	. 2 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → ( bday ‘𝑍) = (dom 𝑆 ∪ suc ∪ ( bday “ 𝐵)))
31		imaundi 5545	. . . . . . 7 ⊢ ( bday “ (𝐴 ∪ 𝐵)) = (( bday “ 𝐴) ∪ ( bday “ 𝐵))
32	31	unieqi 4445	. . . . . 6 ⊢ ∪ ( bday “ (𝐴 ∪ 𝐵)) = ∪ (( bday “ 𝐴) ∪ ( bday “ 𝐵))
33		uniun 4456	. . . . . 6 ⊢ ∪ (( bday “ 𝐴) ∪ ( bday “ 𝐵)) = (∪ ( bday “ 𝐴) ∪ ∪ ( bday “ 𝐵))
34	32, 33	eqtri 2644	. . . . 5 ⊢ ∪ ( bday “ (𝐴 ∪ 𝐵)) = (∪ ( bday “ 𝐴) ∪ ∪ ( bday “ 𝐵))
35		suceq 5790	. . . . 5 ⊢ (∪ ( bday “ (𝐴 ∪ 𝐵)) = (∪ ( bday “ 𝐴) ∪ ∪ ( bday “ 𝐵)) → suc ∪ ( bday “ (𝐴 ∪ 𝐵)) = suc (∪ ( bday “ 𝐴) ∪ ∪ ( bday “ 𝐵)))
36	34, 35	ax-mp 5	. . . 4 ⊢ suc ∪ ( bday “ (𝐴 ∪ 𝐵)) = suc (∪ ( bday “ 𝐴) ∪ ∪ ( bday “ 𝐵))
37		imassrn 5477	. . . . . . 7 ⊢ ( bday “ 𝐴) ⊆ ran bday
38		bdayfo 31828	. . . . . . . 8 ⊢ bday : No –onto→On
39		forn 6118	. . . . . . . 8 ⊢ ( bday : No –onto→On → ran bday = On)
40	38, 39	ax-mp 5	. . . . . . 7 ⊢ ran bday = On
41	37, 40	sseqtri 3637	. . . . . 6 ⊢ ( bday “ 𝐴) ⊆ On
42		ssorduni 6985	. . . . . 6 ⊢ (( bday “ 𝐴) ⊆ On → Ord ∪ ( bday “ 𝐴))
43	41, 42	ax-mp 5	. . . . 5 ⊢ Ord ∪ ( bday “ 𝐴)
44		imassrn 5477	. . . . . . 7 ⊢ ( bday “ 𝐵) ⊆ ran bday
45	44, 40	sseqtri 3637	. . . . . 6 ⊢ ( bday “ 𝐵) ⊆ On
46		ssorduni 6985	. . . . . 6 ⊢ (( bday “ 𝐵) ⊆ On → Ord ∪ ( bday “ 𝐵))
47	45, 46	ax-mp 5	. . . . 5 ⊢ Ord ∪ ( bday “ 𝐵)
48		ordsucun 7025	. . . . 5 ⊢ ((Ord ∪ ( bday “ 𝐴) ∧ Ord ∪ ( bday “ 𝐵)) → suc (∪ ( bday “ 𝐴) ∪ ∪ ( bday “ 𝐵)) = (suc ∪ ( bday “ 𝐴) ∪ suc ∪ ( bday “ 𝐵)))
49	43, 47, 48	mp2an 708	. . . 4 ⊢ suc (∪ ( bday “ 𝐴) ∪ ∪ ( bday “ 𝐵)) = (suc ∪ ( bday “ 𝐴) ∪ suc ∪ ( bday “ 𝐵))
50	36, 49	eqtri 2644	. . 3 ⊢ suc ∪ ( bday “ (𝐴 ∪ 𝐵)) = (suc ∪ ( bday “ 𝐴) ∪ suc ∪ ( bday “ 𝐵))
51	50	a1i 11	. 2 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → suc ∪ ( bday “ (𝐴 ∪ 𝐵)) = (suc ∪ ( bday “ 𝐴) ∪ suc ∪ ( bday “ 𝐵)))
52	9, 30, 51	3sstr4d 3648	1 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → ( bday ‘𝑍) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  {cab 2608   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ⊆ wss 3574  ∅c0 3915  ifcif 4086  {csn 4177  ⟨cop 4183  ∪ cuni 4436   class class class wbr 4653   ↦ cmpt 4729   × cxp 5112  dom cdm 5114  ran crn 5115   ↾ cres 5116   “ cima 5117  Ord word 5722  Oncon0 5723  suc csuc 5725  ℩cio 5849  –onto→wfo 5886  ‘cfv 5888  ℩crio 6610  1_𝑜c1o 7553  2_𝑜c2o 7554   No csur 31793   <s cslt 31794   bday cbday 31795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-1o 7560  df-2o 7561  df-no 31796  df-slt 31797  df-bday 31798

This theorem is referenced by:  noetalem5  31867