Theorem noetalem4 33215

Description: Lemma for noeta 33217. 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 33198	. . . . . 6 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → 𝑆 ∈ No )
3		bdayval 33150	. . . . . 6 ⊢ (𝑆 ∈ No → ( bday ‘𝑆) = dom 𝑆)
4	2, 3	syl 17	. . . . 5 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → ( bday ‘𝑆) = dom 𝑆)
5	1	nosupbday 33200	. . . . 5 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → ( bday ‘𝑆) ⊆ suc ∪ ( bday “ 𝐴))
6	4, 5	eqsstrrd 4005	. . . 4 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → dom 𝑆 ⊆ suc ∪ ( bday “ 𝐴))
7	6	adantr 483	. . 3 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → dom 𝑆 ⊆ suc ∪ ( bday “ 𝐴))
8		unss1 4154	. . 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 765	. . . . 5 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → 𝐴 ⊆ No )
11		simplr 767	. . . . 5 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → 𝐴 ∈ V)
12		simprr 771	. . . . 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 33212	. . . . 5 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝑍 ∈ No )
15	10, 11, 12, 14	syl3anc 1367	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → 𝑍 ∈ No )
16		bdayval 33150	. . . 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 5767	. . . 4 ⊢ dom 𝑍 = dom (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}))
19		dmun 5773	. . . . 5 ⊢ dom (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o})) = (dom 𝑆 ∪ dom ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}))
20		1oex 8104	. . . . . . . . 9 ⊢ 1o ∈ V
21	20	snnz 4704	. . . . . . . 8 ⊢ {1o} ≠ ∅
22		dmxp 5793	. . . . . . . 8 ⊢ ({1o} ≠ ∅ → dom ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) = (suc ∪ ( bday “ 𝐵) ∖ dom 𝑆))
23	21, 22	ax-mp 5	. . . . . . 7 ⊢ dom ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}) = (suc ∪ ( bday “ 𝐵) ∖ dom 𝑆)
24	23	uneq2i 4135	. . . . . 6 ⊢ (dom 𝑆 ∪ dom ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o})) = (dom 𝑆 ∪ (suc ∪ ( bday “ 𝐵) ∖ dom 𝑆))
25		undif2 4424	. . . . . 6 ⊢ (dom 𝑆 ∪ (suc ∪ ( bday “ 𝐵) ∖ dom 𝑆)) = (dom 𝑆 ∪ suc ∪ ( bday “ 𝐵))
26	24, 25	eqtri 2844	. . . . 5 ⊢ (dom 𝑆 ∪ dom ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o})) = (dom 𝑆 ∪ suc ∪ ( bday “ 𝐵))
27	19, 26	eqtri 2844	. . . 4 ⊢ dom (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o})) = (dom 𝑆 ∪ suc ∪ ( bday “ 𝐵))
28	18, 27	eqtri 2844	. . 3 ⊢ dom 𝑍 = (dom 𝑆 ∪ suc ∪ ( bday “ 𝐵))
29	17, 28	syl6eq 2872	. 2 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → ( bday ‘𝑍) = (dom 𝑆 ∪ suc ∪ ( bday “ 𝐵)))
30		imaundi 6002	. . . . . . 7 ⊢ ( bday “ (𝐴 ∪ 𝐵)) = (( bday “ 𝐴) ∪ ( bday “ 𝐵))
31	30	unieqi 4840	. . . . . 6 ⊢ ∪ ( bday “ (𝐴 ∪ 𝐵)) = ∪ (( bday “ 𝐴) ∪ ( bday “ 𝐵))
32		uniun 4850	. . . . . 6 ⊢ ∪ (( bday “ 𝐴) ∪ ( bday “ 𝐵)) = (∪ ( bday “ 𝐴) ∪ ∪ ( bday “ 𝐵))
33	31, 32	eqtri 2844	. . . . 5 ⊢ ∪ ( bday “ (𝐴 ∪ 𝐵)) = (∪ ( bday “ 𝐴) ∪ ∪ ( bday “ 𝐵))
34		suceq 6250	. . . . 5 ⊢ (∪ ( bday “ (𝐴 ∪ 𝐵)) = (∪ ( bday “ 𝐴) ∪ ∪ ( bday “ 𝐵)) → suc ∪ ( bday “ (𝐴 ∪ 𝐵)) = suc (∪ ( bday “ 𝐴) ∪ ∪ ( bday “ 𝐵)))
35	33, 34	ax-mp 5	. . . 4 ⊢ suc ∪ ( bday “ (𝐴 ∪ 𝐵)) = suc (∪ ( bday “ 𝐴) ∪ ∪ ( bday “ 𝐵))
36		imassrn 5934	. . . . . . 7 ⊢ ( bday “ 𝐴) ⊆ ran bday
37		bdayfo 33177	. . . . . . . 8 ⊢ bday : No –onto→On
38		forn 6587	. . . . . . . 8 ⊢ ( bday : No –onto→On → ran bday = On)
39	37, 38	ax-mp 5	. . . . . . 7 ⊢ ran bday = On
40	36, 39	sseqtri 4002	. . . . . 6 ⊢ ( bday “ 𝐴) ⊆ On
41		ssorduni 7494	. . . . . 6 ⊢ (( bday “ 𝐴) ⊆ On → Ord ∪ ( bday “ 𝐴))
42	40, 41	ax-mp 5	. . . . 5 ⊢ Ord ∪ ( bday “ 𝐴)
43		imassrn 5934	. . . . . . 7 ⊢ ( bday “ 𝐵) ⊆ ran bday
44	43, 39	sseqtri 4002	. . . . . 6 ⊢ ( bday “ 𝐵) ⊆ On
45		ssorduni 7494	. . . . . 6 ⊢ (( bday “ 𝐵) ⊆ On → Ord ∪ ( bday “ 𝐵))
46	44, 45	ax-mp 5	. . . . 5 ⊢ Ord ∪ ( bday “ 𝐵)
47		ordsucun 7534	. . . . 5 ⊢ ((Ord ∪ ( bday “ 𝐴) ∧ Ord ∪ ( bday “ 𝐵)) → suc (∪ ( bday “ 𝐴) ∪ ∪ ( bday “ 𝐵)) = (suc ∪ ( bday “ 𝐴) ∪ suc ∪ ( bday “ 𝐵)))
48	42, 46, 47	mp2an 690	. . . 4 ⊢ suc (∪ ( bday “ 𝐴) ∪ ∪ ( bday “ 𝐵)) = (suc ∪ ( bday “ 𝐴) ∪ suc ∪ ( bday “ 𝐵))
49	35, 48	eqtri 2844	. . 3 ⊢ suc ∪ ( bday “ (𝐴 ∪ 𝐵)) = (suc ∪ ( bday “ 𝐴) ∪ suc ∪ ( bday “ 𝐵))
50	49	a1i 11	. 2 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → suc ∪ ( bday “ (𝐴 ∪ 𝐵)) = (suc ∪ ( bday “ 𝐴) ∪ suc ∪ ( bday “ 𝐵)))
51	9, 29, 50	3sstr4d 4013	1 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → ( bday ‘𝑍) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  {cab 2799   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  Vcvv 3494   ∖ cdif 3932   ∪ cun 3933   ⊆ wss 3935  ∅c0 4290  ifcif 4466  {csn 4560  ⟨cop 4566  ∪ cuni 4831   class class class wbr 5058   ↦ cmpt 5138   × cxp 5547  dom cdm 5549  ran crn 5550   ↾ cres 5551   “ cima 5552  Ord word 6184  Oncon0 6185  suc csuc 6187  ℩cio 6306  –onto→wfo 6347  ‘cfv 6349  ℩crio 7107  1_oc1o 8089  2_oc2o 8090   No csur 33142   <s cslt 33143   bday cbday 33144

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-ord 6188  df-on 6189  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-1o 8096  df-2o 8097  df-no 33145  df-slt 33146  df-bday 33147

This theorem is referenced by:  noetalem5  33216