Theorem noeta2 28147

Description: A version of noeta 28100 with fewer hypotheses but a weaker upper bound (Contributed by Scott Fenton, 7-Dec-2021.)

Assertion

Ref	Expression
noeta2	⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → ∃𝑧 ∈ No (∀𝑥 ∈ 𝐴 𝑥 <s 𝑧 ∧ ∀𝑦 ∈ 𝐵 𝑧 <s 𝑦 ∧ (bday‘𝑧) ⊆ suc ∪ (bday “ (𝐴 ∪ 𝐵))))

Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧

Allowed substitution hints:   𝑉(𝑥, 𝑦, 𝑧)   𝑊(𝑥, 𝑦, 𝑧)

Proof of Theorem noeta2

Step	Hyp	Ref	Expression
1		id 23	. 2 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → ((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦))
2		bdayfo 28034	. . . . . . 7 ⊢ bday:No–onto→On
3		fofun 6797	. . . . . . 7 ⊢ (bday:No–onto→On → Fun bday)
4	2, 3	ax-mp 5	. . . . . 6 ⊢ Fun bday
5		simp1r 1217	. . . . . . 7 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → 𝐴 ∈ 𝑉)
6		simp2r 1219	. . . . . . 7 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → 𝐵 ∈ 𝑊)
7		unexg 7760	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
8	5, 6, 7	syl2anc 596	. . . . . 6 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → (𝐴 ∪ 𝐵) ∈ V)
9		funimaexg 6626	. . . . . 6 ⊢ ((Fun bday ∧ (𝐴 ∪ 𝐵) ∈ V) → (bday “ (𝐴 ∪ 𝐵)) ∈ V)
10	4, 8, 9	sylancr 599	. . . . 5 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → (bday “ (𝐴 ∪ 𝐵)) ∈ V)
11	10	uniexd 7759	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → ∪ (bday “ (𝐴 ∪ 𝐵)) ∈ V)
12		imassrn 6197	. . . . . . 7 ⊢ (bday “ (𝐴 ∪ 𝐵)) ⊆ ran bday
13		forn 6799	. . . . . . . 8 ⊢ (bday:No–onto→On → ran bday = On)
14	2, 13	ax-mp 5	. . . . . . 7 ⊢ ran bday = On
15	12, 14	sseqtri 3956	. . . . . 6 ⊢ (bday “ (𝐴 ∪ 𝐵)) ⊆ On
16		ssorduni 7793	. . . . . 6 ⊢ ((bday “ (𝐴 ∪ 𝐵)) ⊆ On → Ord ∪ (bday “ (𝐴 ∪ 𝐵)))
17	15, 16	ax-mp 5	. . . . 5 ⊢ Ord ∪ (bday “ (𝐴 ∪ 𝐵))
18		elon2 6373	. . . . 5 ⊢ (∪ (bday “ (𝐴 ∪ 𝐵)) ∈ On ↔ (Ord ∪ (bday “ (𝐴 ∪ 𝐵)) ∧ ∪ (bday “ (𝐴 ∪ 𝐵)) ∈ V))
19	17, 18	mpbiran 722	. . . 4 ⊢ (∪ (bday “ (𝐴 ∪ 𝐵)) ∈ On ↔ ∪ (bday “ (𝐴 ∪ 𝐵)) ∈ V)
20	11, 19	sylibr 237	. . 3 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → ∪ (bday “ (𝐴 ∪ 𝐵)) ∈ On)
21		onsucb 7828	. . 3 ⊢ (∪ (bday “ (𝐴 ∪ 𝐵)) ∈ On ↔ suc ∪ (bday “ (𝐴 ∪ 𝐵)) ∈ On)
22	20, 21	sylib 221	. 2 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → suc ∪ (bday “ (𝐴 ∪ 𝐵)) ∈ On)
23		onsucuni 7839	. . 3 ⊢ ((bday “ (𝐴 ∪ 𝐵)) ⊆ On → (bday “ (𝐴 ∪ 𝐵)) ⊆ suc ∪ (bday “ (𝐴 ∪ 𝐵)))
24	15, 23	mp1i 14	. 2 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → (bday “ (𝐴 ∪ 𝐵)) ⊆ suc ∪ (bday “ (𝐴 ∪ 𝐵)))
25		noeta 28100	. 2 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) ∧ (suc ∪ (bday “ (𝐴 ∪ 𝐵)) ∈ On ∧ (bday “ (𝐴 ∪ 𝐵)) ⊆ suc ∪ (bday “ (𝐴 ∪ 𝐵)))) → ∃𝑧 ∈ No (∀𝑥 ∈ 𝐴 𝑥 <s 𝑧 ∧ ∀𝑦 ∈ 𝐵 𝑧 <s 𝑦 ∧ (bday‘𝑧) ⊆ suc ∪ (bday “ (𝐴 ∪ 𝐵))))
26	1, 22, 24, 25	syl12anc 850	1 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → ∃𝑧 ∈ No (∀𝑥 ∈ 𝐴 𝑥 <s 𝑧 ∧ ∀𝑦 ∈ 𝐵 𝑧 <s 𝑦 ∧ (bday‘𝑧) ⊆ suc ∪ (bday “ (𝐴 ∪ 𝐵))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3895   ∪ cun 4149  ∪ cuni 4866   class class class wbr 5102  ran crn 5651   “ cima 5653  Ord word 6361  Oncon0 6362  suc csuc 6364  Fun wfun 6532  –onto→wfo 6536  ‘cfv 6538  Nocsur 27997   <s clts 27998  bdaycbday 27999

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  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 7377  df-1o 8476  df-2o 8477  df-no 28000  df-lts 28001  df-bday 28002

This theorem is used by:  conway  28165