Theorem noeta2 27761

Description: A version of noeta 27715 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 22	. 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 27649	. . . . . . 7 ⊢ bday : No –onto→On
3		fofun 6748	. . . . . . 7 ⊢ ( bday : No –onto→On → Fun bday )
4	2, 3	ax-mp 5	. . . . . 6 ⊢ Fun bday
5		simp1r 1200	. . . . . . 7 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → 𝐴 ∈ 𝑉)
6		simp2r 1202	. . . . . . 7 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → 𝐵 ∈ 𝑊)
7		unexg 7690	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
8	5, 6, 7	syl2anc 585	. . . . . 6 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → (𝐴 ∪ 𝐵) ∈ V)
9		funimaexg 6580	. . . . . 6 ⊢ ((Fun bday ∧ (𝐴 ∪ 𝐵) ∈ V) → ( bday “ (𝐴 ∪ 𝐵)) ∈ V)
10	4, 8, 9	sylancr 588	. . . . 5 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → ( bday “ (𝐴 ∪ 𝐵)) ∈ V)
11	10	uniexd 7689	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ V)
12		imassrn 6031	. . . . . . 7 ⊢ ( bday “ (𝐴 ∪ 𝐵)) ⊆ ran bday
13		forn 6750	. . . . . . . 8 ⊢ ( bday : No –onto→On → ran bday = On)
14	2, 13	ax-mp 5	. . . . . . 7 ⊢ ran bday = On
15	12, 14	sseqtri 3983	. . . . . 6 ⊢ ( bday “ (𝐴 ∪ 𝐵)) ⊆ On
16		ssorduni 7726	. . . . . 6 ⊢ (( bday “ (𝐴 ∪ 𝐵)) ⊆ On → Ord ∪ ( bday “ (𝐴 ∪ 𝐵)))
17	15, 16	ax-mp 5	. . . . 5 ⊢ Ord ∪ ( bday “ (𝐴 ∪ 𝐵))
18		elon2 6329	. . . . 5 ⊢ (∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ On ↔ (Ord ∪ ( bday “ (𝐴 ∪ 𝐵)) ∧ ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ V))
19	17, 18	mpbiran 710	. . . 4 ⊢ (∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ On ↔ ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ V)
20	11, 19	sylibr 234	. . 3 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ On)
21		onsucb 7761	. . 3 ⊢ (∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ On ↔ suc ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ On)
22	20, 21	sylib 218	. 2 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → suc ∪ ( bday “ (𝐴 ∪ 𝐵)) ∈ On)
23		onsucuni 7772	. . 3 ⊢ (( bday “ (𝐴 ∪ 𝐵)) ⊆ On → ( bday “ (𝐴 ∪ 𝐵)) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))
24	15, 23	mp1i 13	. 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 27715	. 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 837	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))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3061  Vcvv 3441   ∪ cun 3900   ⊆ wss 3902  ∪ cuni 4864   class class class wbr 5099  ran crn 5626   “ cima 5628  Ord word 6317  Oncon0 6318  suc csuc 6320  Fun wfun 6487  –onto→wfo 6491  ‘cfv 6493   No csur 27611   <s clts 27612   bday cbday 27613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-int 4904  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-1o 8399  df-2o 8400  df-no 27614  df-lts 27615  df-bday 27616

This theorem is referenced by:  conway  27779