Theorem nofulllem4 31104

Description: Lemma for nofull (future) . Show a particular abstraction is an ordinal. (Contributed by Scott Fenton, 25-Apr-2017.)

Hypothesis

Ref	Expression
nofulllem4.1	⊢ 𝑀 = ∩ {𝑎 ∈ On ∣ ∀𝑥 ∈ 𝐿 ∀𝑦 ∈ 𝑅 (𝑥 ↾ 𝑎) ≠ (𝑦 ↾ 𝑎)}

Assertion

Ref	Expression
nofulllem4	⊢ (((𝐿 ⊆ No ∧ 𝐿 ∈ 𝑉) ∧ (𝑅 ⊆ No ∧ 𝑅 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐿 ∀𝑦 ∈ 𝑅 𝑥 <s 𝑦) → 𝑀 ∈ On)

Distinct variable groups:   𝑥,𝐿,𝑦,𝑎   𝑥,𝑅,𝑦,𝑎   𝑥,𝑊,𝑦   𝑥,𝑉,𝑦

Allowed substitution hints:   𝑀(𝑥,𝑦,𝑎)   𝑉(𝑎)   𝑊(𝑎)

Proof of Theorem nofulllem4

Step	Hyp	Ref	Expression
1		nofulllem4.1	. 2 ⊢ 𝑀 = ∩ {𝑎 ∈ On ∣ ∀𝑥 ∈ 𝐿 ∀𝑦 ∈ 𝑅 (𝑥 ↾ 𝑎) ≠ (𝑦 ↾ 𝑎)}
2		unexg 6857	. . . . . . 7 ⊢ ((𝐿 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝐿 ∪ 𝑅) ∈ V)
3		nobndlem1 31091	. . . . . . . 8 ⊢ ((𝐿 ∪ 𝑅) ∈ V → suc ∪ ( bday “ (𝐿 ∪ 𝑅)) ∈ On)
4		sucelon 6909	. . . . . . . 8 ⊢ (∪ ( bday “ (𝐿 ∪ 𝑅)) ∈ On ↔ suc ∪ ( bday “ (𝐿 ∪ 𝑅)) ∈ On)
5	3, 4	sylibr 223	. . . . . . 7 ⊢ ((𝐿 ∪ 𝑅) ∈ V → ∪ ( bday “ (𝐿 ∪ 𝑅)) ∈ On)
6	2, 5	syl 17	. . . . . 6 ⊢ ((𝐿 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ∪ ( bday “ (𝐿 ∪ 𝑅)) ∈ On)
7	6	ad2ant2l 778	. . . . 5 ⊢ (((𝐿 ⊆ No ∧ 𝐿 ∈ 𝑉) ∧ (𝑅 ⊆ No ∧ 𝑅 ∈ 𝑊)) → ∪ ( bday “ (𝐿 ∪ 𝑅)) ∈ On)
8	7	3adant3 1074	. . . 4 ⊢ (((𝐿 ⊆ No ∧ 𝐿 ∈ 𝑉) ∧ (𝑅 ⊆ No ∧ 𝑅 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐿 ∀𝑦 ∈ 𝑅 𝑥 <s 𝑦) → ∪ ( bday “ (𝐿 ∪ 𝑅)) ∈ On)
9		simprl 790	. . . . . . . . . . 11 ⊢ (((𝐿 ⊆ No ∧ 𝐿 ∈ 𝑉) ∧ (𝑅 ⊆ No ∧ 𝑅 ∈ 𝑊)) → 𝑅 ⊆ No )
10		simpr 476	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑅) → 𝑦 ∈ 𝑅)
11		ssel2 3563	. . . . . . . . . . 11 ⊢ ((𝑅 ⊆ No ∧ 𝑦 ∈ 𝑅) → 𝑦 ∈ No )
12	9, 10, 11	syl2an 493	. . . . . . . . . 10 ⊢ ((((𝐿 ⊆ No ∧ 𝐿 ∈ 𝑉) ∧ (𝑅 ⊆ No ∧ 𝑅 ∈ 𝑊)) ∧ (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑅)) → 𝑦 ∈ No )
13		sltirr 31069	. . . . . . . . . 10 ⊢ (𝑦 ∈ No → ¬ 𝑦 <s 𝑦)
14		breq1 4586	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑥 <s 𝑦 ↔ 𝑦 <s 𝑦))
15	14	notbid 307	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 <s 𝑦 ↔ ¬ 𝑦 <s 𝑦))
16	15	biimprcd 239	. . . . . . . . . . 11 ⊢ (¬ 𝑦 <s 𝑦 → (𝑥 = 𝑦 → ¬ 𝑥 <s 𝑦))
17	16	necon2ad 2797	. . . . . . . . . 10 ⊢ (¬ 𝑦 <s 𝑦 → (𝑥 <s 𝑦 → 𝑥 ≠ 𝑦))
18	12, 13, 17	3syl 18	. . . . . . . . 9 ⊢ ((((𝐿 ⊆ No ∧ 𝐿 ∈ 𝑉) ∧ (𝑅 ⊆ No ∧ 𝑅 ∈ 𝑊)) ∧ (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑅)) → (𝑥 <s 𝑦 → 𝑥 ≠ 𝑦))
19	18	impr 647	. . . . . . . 8 ⊢ ((((𝐿 ⊆ No ∧ 𝐿 ∈ 𝑉) ∧ (𝑅 ⊆ No ∧ 𝑅 ∈ 𝑊)) ∧ ((𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑅) ∧ 𝑥 <s 𝑦)) → 𝑥 ≠ 𝑦)
20		simpll 786	. . . . . . . . 9 ⊢ (((𝐿 ⊆ No ∧ 𝐿 ∈ 𝑉) ∧ (𝑅 ⊆ No ∧ 𝑅 ∈ 𝑊)) → 𝐿 ⊆ No )
21		simpll 786	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑅) ∧ 𝑥 <s 𝑦) → 𝑥 ∈ 𝐿)
22		ssun1 3738	. . . . . . . . . 10 ⊢ 𝐿 ⊆ (𝐿 ∪ 𝑅)
23		nofulllem3 31103	. . . . . . . . . 10 ⊢ ((𝐿 ⊆ No ∧ 𝑥 ∈ 𝐿 ∧ 𝐿 ⊆ (𝐿 ∪ 𝑅)) → (𝑥 ↾ ∪ ( bday “ (𝐿 ∪ 𝑅))) = 𝑥)
24	22, 23	mp3an3 1405	. . . . . . . . 9 ⊢ ((𝐿 ⊆ No ∧ 𝑥 ∈ 𝐿) → (𝑥 ↾ ∪ ( bday “ (𝐿 ∪ 𝑅))) = 𝑥)
25	20, 21, 24	syl2an 493	. . . . . . . 8 ⊢ ((((𝐿 ⊆ No ∧ 𝐿 ∈ 𝑉) ∧ (𝑅 ⊆ No ∧ 𝑅 ∈ 𝑊)) ∧ ((𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑅) ∧ 𝑥 <s 𝑦)) → (𝑥 ↾ ∪ ( bday “ (𝐿 ∪ 𝑅))) = 𝑥)
26		simplr 788	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑅) ∧ 𝑥 <s 𝑦) → 𝑦 ∈ 𝑅)
27		ssun2 3739	. . . . . . . . . 10 ⊢ 𝑅 ⊆ (𝐿 ∪ 𝑅)
28		nofulllem3 31103	. . . . . . . . . 10 ⊢ ((𝑅 ⊆ No ∧ 𝑦 ∈ 𝑅 ∧ 𝑅 ⊆ (𝐿 ∪ 𝑅)) → (𝑦 ↾ ∪ ( bday “ (𝐿 ∪ 𝑅))) = 𝑦)
29	27, 28	mp3an3 1405	. . . . . . . . 9 ⊢ ((𝑅 ⊆ No ∧ 𝑦 ∈ 𝑅) → (𝑦 ↾ ∪ ( bday “ (𝐿 ∪ 𝑅))) = 𝑦)
30	9, 26, 29	syl2an 493	. . . . . . . 8 ⊢ ((((𝐿 ⊆ No ∧ 𝐿 ∈ 𝑉) ∧ (𝑅 ⊆ No ∧ 𝑅 ∈ 𝑊)) ∧ ((𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑅) ∧ 𝑥 <s 𝑦)) → (𝑦 ↾ ∪ ( bday “ (𝐿 ∪ 𝑅))) = 𝑦)
31	19, 25, 30	3netr4d 2859	. . . . . . 7 ⊢ ((((𝐿 ⊆ No ∧ 𝐿 ∈ 𝑉) ∧ (𝑅 ⊆ No ∧ 𝑅 ∈ 𝑊)) ∧ ((𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑅) ∧ 𝑥 <s 𝑦)) → (𝑥 ↾ ∪ ( bday “ (𝐿 ∪ 𝑅))) ≠ (𝑦 ↾ ∪ ( bday “ (𝐿 ∪ 𝑅))))
32	31	expr 641	. . . . . 6 ⊢ ((((𝐿 ⊆ No ∧ 𝐿 ∈ 𝑉) ∧ (𝑅 ⊆ No ∧ 𝑅 ∈ 𝑊)) ∧ (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑅)) → (𝑥 <s 𝑦 → (𝑥 ↾ ∪ ( bday “ (𝐿 ∪ 𝑅))) ≠ (𝑦 ↾ ∪ ( bday “ (𝐿 ∪ 𝑅)))))
33	32	ralimdvva 2947	. . . . 5 ⊢ (((𝐿 ⊆ No ∧ 𝐿 ∈ 𝑉) ∧ (𝑅 ⊆ No ∧ 𝑅 ∈ 𝑊)) → (∀𝑥 ∈ 𝐿 ∀𝑦 ∈ 𝑅 𝑥 <s 𝑦 → ∀𝑥 ∈ 𝐿 ∀𝑦 ∈ 𝑅 (𝑥 ↾ ∪ ( bday “ (𝐿 ∪ 𝑅))) ≠ (𝑦 ↾ ∪ ( bday “ (𝐿 ∪ 𝑅)))))
34	33	3impia 1253	. . . 4 ⊢ (((𝐿 ⊆ No ∧ 𝐿 ∈ 𝑉) ∧ (𝑅 ⊆ No ∧ 𝑅 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐿 ∀𝑦 ∈ 𝑅 𝑥 <s 𝑦) → ∀𝑥 ∈ 𝐿 ∀𝑦 ∈ 𝑅 (𝑥 ↾ ∪ ( bday “ (𝐿 ∪ 𝑅))) ≠ (𝑦 ↾ ∪ ( bday “ (𝐿 ∪ 𝑅))))
35		reseq2 5312	. . . . . . 7 ⊢ (𝑎 = ∪ ( bday “ (𝐿 ∪ 𝑅)) → (𝑥 ↾ 𝑎) = (𝑥 ↾ ∪ ( bday “ (𝐿 ∪ 𝑅))))
36		reseq2 5312	. . . . . . 7 ⊢ (𝑎 = ∪ ( bday “ (𝐿 ∪ 𝑅)) → (𝑦 ↾ 𝑎) = (𝑦 ↾ ∪ ( bday “ (𝐿 ∪ 𝑅))))
37	35, 36	neeq12d 2843	. . . . . 6 ⊢ (𝑎 = ∪ ( bday “ (𝐿 ∪ 𝑅)) → ((𝑥 ↾ 𝑎) ≠ (𝑦 ↾ 𝑎) ↔ (𝑥 ↾ ∪ ( bday “ (𝐿 ∪ 𝑅))) ≠ (𝑦 ↾ ∪ ( bday “ (𝐿 ∪ 𝑅)))))
38	37	2ralbidv 2972	. . . . 5 ⊢ (𝑎 = ∪ ( bday “ (𝐿 ∪ 𝑅)) → (∀𝑥 ∈ 𝐿 ∀𝑦 ∈ 𝑅 (𝑥 ↾ 𝑎) ≠ (𝑦 ↾ 𝑎) ↔ ∀𝑥 ∈ 𝐿 ∀𝑦 ∈ 𝑅 (𝑥 ↾ ∪ ( bday “ (𝐿 ∪ 𝑅))) ≠ (𝑦 ↾ ∪ ( bday “ (𝐿 ∪ 𝑅)))))
39	38	rspcev 3282	. . . 4 ⊢ ((∪ ( bday “ (𝐿 ∪ 𝑅)) ∈ On ∧ ∀𝑥 ∈ 𝐿 ∀𝑦 ∈ 𝑅 (𝑥 ↾ ∪ ( bday “ (𝐿 ∪ 𝑅))) ≠ (𝑦 ↾ ∪ ( bday “ (𝐿 ∪ 𝑅)))) → ∃𝑎 ∈ On ∀𝑥 ∈ 𝐿 ∀𝑦 ∈ 𝑅 (𝑥 ↾ 𝑎) ≠ (𝑦 ↾ 𝑎))
40	8, 34, 39	syl2anc 691	. . 3 ⊢ (((𝐿 ⊆ No ∧ 𝐿 ∈ 𝑉) ∧ (𝑅 ⊆ No ∧ 𝑅 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐿 ∀𝑦 ∈ 𝑅 𝑥 <s 𝑦) → ∃𝑎 ∈ On ∀𝑥 ∈ 𝐿 ∀𝑦 ∈ 𝑅 (𝑥 ↾ 𝑎) ≠ (𝑦 ↾ 𝑎))
41		onintrab2 6894	. . 3 ⊢ (∃𝑎 ∈ On ∀𝑥 ∈ 𝐿 ∀𝑦 ∈ 𝑅 (𝑥 ↾ 𝑎) ≠ (𝑦 ↾ 𝑎) ↔ ∩ {𝑎 ∈ On ∣ ∀𝑥 ∈ 𝐿 ∀𝑦 ∈ 𝑅 (𝑥 ↾ 𝑎) ≠ (𝑦 ↾ 𝑎)} ∈ On)
42	40, 41	sylib 207	. 2 ⊢ (((𝐿 ⊆ No ∧ 𝐿 ∈ 𝑉) ∧ (𝑅 ⊆ No ∧ 𝑅 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐿 ∀𝑦 ∈ 𝑅 𝑥 <s 𝑦) → ∩ {𝑎 ∈ On ∣ ∀𝑥 ∈ 𝐿 ∀𝑦 ∈ 𝑅 (𝑥 ↾ 𝑎) ≠ (𝑦 ↾ 𝑎)} ∈ On)
43	1, 42	syl5eqel 2692	1 ⊢ (((𝐿 ⊆ No ∧ 𝐿 ∈ 𝑉) ∧ (𝑅 ⊆ No ∧ 𝑅 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐿 ∀𝑦 ∈ 𝑅 𝑥 <s 𝑦) → 𝑀 ∈ On)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∪ cun 3538   ⊆ wss 3540  ∪ cuni 4372  ∩ cint 4410   class class class wbr 4583   ↾ cres 5040   “ cima 5041  Oncon0 5640  suc csuc 5642   No csur 31037   <s cslt 31038   bday cbday 31039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-1o 7447  df-2o 7448  df-no 31040  df-slt 31041  df-bday 31042

This theorem is referenced by:  nofulllem5  31105