Theorem noeta2 28174

Description: A version of noeta 28127 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		bdayfun 28160	. . . . . 6 ⊢ Fun bday
3		simp1r 1217	. . . . . . 7 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → 𝐴 ∈ 𝑉)
4		simp2r 1219	. . . . . . 7 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → 𝐵 ∈ 𝑊)
5	3, 4	unexd 7789	. . . . . 6 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → (𝐴 ∪ 𝐵) ∈ V)
6		funimaexg 6645	. . . . . 6 ⊢ ((Fun bday ∧ (𝐴 ∪ 𝐵) ∈ V) → (bday “ (𝐴 ∪ 𝐵)) ∈ V)
7	2, 5, 6	sylancr 599	. . . . 5 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → (bday “ (𝐴 ∪ 𝐵)) ∈ V)
8	7	uniexd 7780	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → ∪ (bday “ (𝐴 ∪ 𝐵)) ∈ V)
9		imassrn 6209	. . . . . . 7 ⊢ (bday “ (𝐴 ∪ 𝐵)) ⊆ ran bday
10		bdayrn 28164	. . . . . . 7 ⊢ ran bday = On
11	9, 10	sseqtri 3961	. . . . . 6 ⊢ (bday “ (𝐴 ∪ 𝐵)) ⊆ On
12		ssorduni 7814	. . . . . 6 ⊢ ((bday “ (𝐴 ∪ 𝐵)) ⊆ On → Ord ∪ (bday “ (𝐴 ∪ 𝐵)))
13	11, 12	ax-mp 5	. . . . 5 ⊢ Ord ∪ (bday “ (𝐴 ∪ 𝐵))
14		elon2 6388	. . . . 5 ⊢ (∪ (bday “ (𝐴 ∪ 𝐵)) ∈ On ↔ (Ord ∪ (bday “ (𝐴 ∪ 𝐵)) ∧ ∪ (bday “ (𝐴 ∪ 𝐵)) ∈ V))
15	13, 14	mpbiran 722	. . . 4 ⊢ (∪ (bday “ (𝐴 ∪ 𝐵)) ∈ On ↔ ∪ (bday “ (𝐴 ∪ 𝐵)) ∈ V)
16	8, 15	sylibr 237	. . 3 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → ∪ (bday “ (𝐴 ∪ 𝐵)) ∈ On)
17		onsucb 7849	. . 3 ⊢ (∪ (bday “ (𝐴 ∪ 𝐵)) ∈ On ↔ suc ∪ (bday “ (𝐴 ∪ 𝐵)) ∈ On)
18	16, 17	sylib 221	. 2 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → suc ∪ (bday “ (𝐴 ∪ 𝐵)) ∈ On)
19		onsucuni 7860	. . 3 ⊢ ((bday “ (𝐴 ∪ 𝐵)) ⊆ On → (bday “ (𝐴 ∪ 𝐵)) ⊆ suc ∪ (bday “ (𝐴 ∪ 𝐵)))
20	11, 19	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)))
21		noeta 28127	. 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))))
22	1, 18, 20, 21	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   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∪ cun 4152  ∪ cuni 4870   class class class wbr 5106  ran crn 5656   “ cima 5658  Ord word 6376  Oncon0 6377  suc csuc 6379  Fun wfun 6549  ‘cfv 6555  Nocsur 28024   <s clts 28025  bdaycbday 28026

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6380  df-on 6381  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-1o 8501  df-2o 8502  df-no 28027  df-lts 28028  df-bday 28029

This theorem is used by:  conway  28192