Theorem noetalem4 32829

Description: Lemma for noeta 32831. 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 32812	. . . . . 6 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → 𝑆 ∈ No )
3		bdayval 32764	. . . . . 6 ⊢ (𝑆 ∈ No → ( bday ‘𝑆) = dom 𝑆)
4	2, 3	syl 17	. . . . 5 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → ( bday ‘𝑆) = dom 𝑆)
5	1	nosupbday 32814	. . . . 5 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → ( bday ‘𝑆) ⊆ suc ∪ ( bday “ 𝐴))
6	4, 5	eqsstrrd 3927	. . . 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 4076	. . 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 763	. . . . 5 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → 𝐴 ⊆ No )
11		simplr 765	. . . . 5 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → 𝐴 ∈ V)
12		simprr 769	. . . . 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 32826	. . . . 5 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝑍 ∈ No )
15	10, 11, 12, 14	syl3anc 1364	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → 𝑍 ∈ No )
16		bdayval 32764	. . . 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 5659	. . . 4 ⊢ dom 𝑍 = dom (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}))
19		dmun 5665	. . . . 5 ⊢ dom (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o})) = (dom 𝑆 ∪ dom ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}))
20		1oex 7961	. . . . . . . . 9 ⊢ 1o ∈ V
21	20	snnz 4618	. . . . . . . 8 ⊢ {1o} ≠ ∅
22		dmxp 5681	. . . . . . . 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 4057	. . . . . 6 ⊢ (dom 𝑆 ∪ dom ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o})) = (dom 𝑆 ∪ (suc ∪ ( bday “ 𝐵) ∖ dom 𝑆))
25		undif2 4339	. . . . . 6 ⊢ (dom 𝑆 ∪ (suc ∪ ( bday “ 𝐵) ∖ dom 𝑆)) = (dom 𝑆 ∪ suc ∪ ( bday “ 𝐵))
26	24, 25	eqtri 2819	. . . . 5 ⊢ (dom 𝑆 ∪ dom ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o})) = (dom 𝑆 ∪ suc ∪ ( bday “ 𝐵))
27	19, 26	eqtri 2819	. . . 4 ⊢ dom (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o})) = (dom 𝑆 ∪ suc ∪ ( bday “ 𝐵))
28	18, 27	eqtri 2819	. . 3 ⊢ dom 𝑍 = (dom 𝑆 ∪ suc ∪ ( bday “ 𝐵))
29	17, 28	syl6eq 2847	. 2 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → ( bday ‘𝑍) = (dom 𝑆 ∪ suc ∪ ( bday “ 𝐵)))
30		imaundi 5884	. . . . . . 7 ⊢ ( bday “ (𝐴 ∪ 𝐵)) = (( bday “ 𝐴) ∪ ( bday “ 𝐵))
31	30	unieqi 4754	. . . . . 6 ⊢ ∪ ( bday “ (𝐴 ∪ 𝐵)) = ∪ (( bday “ 𝐴) ∪ ( bday “ 𝐵))
32		uniun 4764	. . . . . 6 ⊢ ∪ (( bday “ 𝐴) ∪ ( bday “ 𝐵)) = (∪ ( bday “ 𝐴) ∪ ∪ ( bday “ 𝐵))
33	31, 32	eqtri 2819	. . . . 5 ⊢ ∪ ( bday “ (𝐴 ∪ 𝐵)) = (∪ ( bday “ 𝐴) ∪ ∪ ( bday “ 𝐵))
34		suceq 6131	. . . . 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 5817	. . . . . . 7 ⊢ ( bday “ 𝐴) ⊆ ran bday
37		bdayfo 32791	. . . . . . . 8 ⊢ bday : No –onto→On
38		forn 6461	. . . . . . . 8 ⊢ ( bday : No –onto→On → ran bday = On)
39	37, 38	ax-mp 5	. . . . . . 7 ⊢ ran bday = On
40	36, 39	sseqtri 3924	. . . . . 6 ⊢ ( bday “ 𝐴) ⊆ On
41		ssorduni 7356	. . . . . 6 ⊢ (( bday “ 𝐴) ⊆ On → Ord ∪ ( bday “ 𝐴))
42	40, 41	ax-mp 5	. . . . 5 ⊢ Ord ∪ ( bday “ 𝐴)
43		imassrn 5817	. . . . . . 7 ⊢ ( bday “ 𝐵) ⊆ ran bday
44	43, 39	sseqtri 3924	. . . . . 6 ⊢ ( bday “ 𝐵) ⊆ On
45		ssorduni 7356	. . . . . 6 ⊢ (( bday “ 𝐵) ⊆ On → Ord ∪ ( bday “ 𝐵))
46	44, 45	ax-mp 5	. . . . 5 ⊢ Ord ∪ ( bday “ 𝐵)
47		ordsucun 7396	. . . . 5 ⊢ ((Ord ∪ ( bday “ 𝐴) ∧ Ord ∪ ( bday “ 𝐵)) → suc (∪ ( bday “ 𝐴) ∪ ∪ ( bday “ 𝐵)) = (suc ∪ ( bday “ 𝐴) ∪ suc ∪ ( bday “ 𝐵)))
48	42, 46, 47	mp2an 688	. . . 4 ⊢ suc (∪ ( bday “ 𝐴) ∪ ∪ ( bday “ 𝐵)) = (suc ∪ ( bday “ 𝐴) ∪ suc ∪ ( bday “ 𝐵))
49	35, 48	eqtri 2819	. . 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 3935	1 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → ( bday ‘𝑍) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081  {cab 2775   ≠ wne 2984  ∀wral 3105  ∃wrex 3106  Vcvv 3437   ∖ cdif 3856   ∪ cun 3857   ⊆ wss 3859  ∅c0 4211  ifcif 4381  {csn 4472  ⟨cop 4478  ∪ cuni 4745   class class class wbr 4962   ↦ cmpt 5041   × cxp 5441  dom cdm 5443  ran crn 5444   ↾ cres 5445   “ cima 5446  Ord word 6065  Oncon0 6066  suc csuc 6068  ℩cio 6187  –onto→wfo 6223  ‘cfv 6225  ℩crio 6976  1_oc1o 7946  2_oc2o 7947   No csur 32756   <s cslt 32757   bday cbday 32758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-ord 6069  df-on 6070  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-1o 7953  df-2o 7954  df-no 32759  df-slt 32760  df-bday 32761

This theorem is referenced by:  noetalem5  32830