Theorem nobndlem2 31092

Description: Lemma for nobndup 31099 and nobnddown 31100. Show a particular abstraction is an ordinal. (Contributed by Scott Fenton, 17-Aug-2011.)

Hypotheses

Ref	Expression
nobndlem2.1	⊢ 𝑋 ∈ {1𝑜, 2𝑜}
nobndlem2.2	⊢ 𝐶 = ∩ {𝑎 ∈ On ∣ ∀𝑛 ∈ 𝐹 ∃𝑏 ∈ 𝑎 (𝑛‘𝑏) ≠ 𝑋}

Assertion

Ref	Expression
nobndlem2	⊢ ((𝐹 ⊆ No ∧ 𝐹 ∈ 𝐴) → 𝐶 ∈ On)

Distinct variable groups:   𝐹,𝑎,𝑏,𝑛   𝑋,𝑎,𝑏

Allowed substitution hints:   𝐴(𝑛,𝑎,𝑏)   𝐶(𝑛,𝑎,𝑏)   𝑋(𝑛)

Proof of Theorem nobndlem2

Step	Hyp	Ref	Expression
1		nobndlem2.2	. 2 ⊢ 𝐶 = ∩ {𝑎 ∈ On ∣ ∀𝑛 ∈ 𝐹 ∃𝑏 ∈ 𝑎 (𝑛‘𝑏) ≠ 𝑋}
2		nobndlem1 31091	. . . 4 ⊢ (𝐹 ∈ 𝐴 → suc ∪ ( bday “ 𝐹) ∈ On)
3		ssel2 3563	. . . . . . . . . 10 ⊢ ((𝐹 ⊆ No ∧ 𝑛 ∈ 𝐹) → 𝑛 ∈ No )
4		bdaydm 31077	. . . . . . . . . 10 ⊢ dom bday = No
5	3, 4	syl6eleqr 2699	. . . . . . . . 9 ⊢ ((𝐹 ⊆ No ∧ 𝑛 ∈ 𝐹) → 𝑛 ∈ dom bday )
6		simpr 476	. . . . . . . . 9 ⊢ ((𝐹 ⊆ No ∧ 𝑛 ∈ 𝐹) → 𝑛 ∈ 𝐹)
7		bdayfun 31075	. . . . . . . . . 10 ⊢ Fun bday
8		funfvima 6396	. . . . . . . . . 10 ⊢ ((Fun bday ∧ 𝑛 ∈ dom bday ) → (𝑛 ∈ 𝐹 → ( bday ‘𝑛) ∈ ( bday “ 𝐹)))
9	7, 8	mpan 702	. . . . . . . . 9 ⊢ (𝑛 ∈ dom bday → (𝑛 ∈ 𝐹 → ( bday ‘𝑛) ∈ ( bday “ 𝐹)))
10	5, 6, 9	sylc 63	. . . . . . . 8 ⊢ ((𝐹 ⊆ No ∧ 𝑛 ∈ 𝐹) → ( bday ‘𝑛) ∈ ( bday “ 𝐹))
11		elssuni 4403	. . . . . . . 8 ⊢ (( bday ‘𝑛) ∈ ( bday “ 𝐹) → ( bday ‘𝑛) ⊆ ∪ ( bday “ 𝐹))
12	10, 11	syl 17	. . . . . . 7 ⊢ ((𝐹 ⊆ No ∧ 𝑛 ∈ 𝐹) → ( bday ‘𝑛) ⊆ ∪ ( bday “ 𝐹))
13		bdayelon 31079	. . . . . . . 8 ⊢ ( bday ‘𝑛) ∈ On
14		imassrn 5396	. . . . . . . . . 10 ⊢ ( bday “ 𝐹) ⊆ ran bday
15		bdayrn 31076	. . . . . . . . . 10 ⊢ ran bday = On
16	14, 15	sseqtri 3600	. . . . . . . . 9 ⊢ ( bday “ 𝐹) ⊆ On
17		ssorduni 6877	. . . . . . . . 9 ⊢ (( bday “ 𝐹) ⊆ On → Ord ∪ ( bday “ 𝐹))
18	16, 17	ax-mp 5	. . . . . . . 8 ⊢ Ord ∪ ( bday “ 𝐹)
19		ordsssuc 5729	. . . . . . . 8 ⊢ ((( bday ‘𝑛) ∈ On ∧ Ord ∪ ( bday “ 𝐹)) → (( bday ‘𝑛) ⊆ ∪ ( bday “ 𝐹) ↔ ( bday ‘𝑛) ∈ suc ∪ ( bday “ 𝐹)))
20	13, 18, 19	mp2an 704	. . . . . . 7 ⊢ (( bday ‘𝑛) ⊆ ∪ ( bday “ 𝐹) ↔ ( bday ‘𝑛) ∈ suc ∪ ( bday “ 𝐹))
21	12, 20	sylib 207	. . . . . 6 ⊢ ((𝐹 ⊆ No ∧ 𝑛 ∈ 𝐹) → ( bday ‘𝑛) ∈ suc ∪ ( bday “ 𝐹))
22		nobndlem2.1	. . . . . . . 8 ⊢ 𝑋 ∈ {1𝑜, 2𝑜}
23	22	nosgnn0i 31056	. . . . . . 7 ⊢ ∅ ≠ 𝑋
24		fvnobday 31081	. . . . . . . . 9 ⊢ (𝑛 ∈ No → (𝑛‘( bday ‘𝑛)) = ∅)
25	3, 24	syl 17	. . . . . . . 8 ⊢ ((𝐹 ⊆ No ∧ 𝑛 ∈ 𝐹) → (𝑛‘( bday ‘𝑛)) = ∅)
26	25	neeq1d 2841	. . . . . . 7 ⊢ ((𝐹 ⊆ No ∧ 𝑛 ∈ 𝐹) → ((𝑛‘( bday ‘𝑛)) ≠ 𝑋 ↔ ∅ ≠ 𝑋))
27	23, 26	mpbiri 247	. . . . . 6 ⊢ ((𝐹 ⊆ No ∧ 𝑛 ∈ 𝐹) → (𝑛‘( bday ‘𝑛)) ≠ 𝑋)
28		fveq2 6103	. . . . . . . 8 ⊢ (𝑏 = ( bday ‘𝑛) → (𝑛‘𝑏) = (𝑛‘( bday ‘𝑛)))
29	28	neeq1d 2841	. . . . . . 7 ⊢ (𝑏 = ( bday ‘𝑛) → ((𝑛‘𝑏) ≠ 𝑋 ↔ (𝑛‘( bday ‘𝑛)) ≠ 𝑋))
30	29	rspcev 3282	. . . . . 6 ⊢ ((( bday ‘𝑛) ∈ suc ∪ ( bday “ 𝐹) ∧ (𝑛‘( bday ‘𝑛)) ≠ 𝑋) → ∃𝑏 ∈ suc ∪ ( bday “ 𝐹)(𝑛‘𝑏) ≠ 𝑋)
31	21, 27, 30	syl2anc 691	. . . . 5 ⊢ ((𝐹 ⊆ No ∧ 𝑛 ∈ 𝐹) → ∃𝑏 ∈ suc ∪ ( bday “ 𝐹)(𝑛‘𝑏) ≠ 𝑋)
32	31	ralrimiva 2949	. . . 4 ⊢ (𝐹 ⊆ No → ∀𝑛 ∈ 𝐹 ∃𝑏 ∈ suc ∪ ( bday “ 𝐹)(𝑛‘𝑏) ≠ 𝑋)
33		rexeq 3116	. . . . . 6 ⊢ (𝑎 = suc ∪ ( bday “ 𝐹) → (∃𝑏 ∈ 𝑎 (𝑛‘𝑏) ≠ 𝑋 ↔ ∃𝑏 ∈ suc ∪ ( bday “ 𝐹)(𝑛‘𝑏) ≠ 𝑋))
34	33	ralbidv 2969	. . . . 5 ⊢ (𝑎 = suc ∪ ( bday “ 𝐹) → (∀𝑛 ∈ 𝐹 ∃𝑏 ∈ 𝑎 (𝑛‘𝑏) ≠ 𝑋 ↔ ∀𝑛 ∈ 𝐹 ∃𝑏 ∈ suc ∪ ( bday “ 𝐹)(𝑛‘𝑏) ≠ 𝑋))
35	34	rspcev 3282	. . . 4 ⊢ ((suc ∪ ( bday “ 𝐹) ∈ On ∧ ∀𝑛 ∈ 𝐹 ∃𝑏 ∈ suc ∪ ( bday “ 𝐹)(𝑛‘𝑏) ≠ 𝑋) → ∃𝑎 ∈ On ∀𝑛 ∈ 𝐹 ∃𝑏 ∈ 𝑎 (𝑛‘𝑏) ≠ 𝑋)
36	2, 32, 35	syl2anr 494	. . 3 ⊢ ((𝐹 ⊆ No ∧ 𝐹 ∈ 𝐴) → ∃𝑎 ∈ On ∀𝑛 ∈ 𝐹 ∃𝑏 ∈ 𝑎 (𝑛‘𝑏) ≠ 𝑋)
37		onintrab2 6894	. . 3 ⊢ (∃𝑎 ∈ On ∀𝑛 ∈ 𝐹 ∃𝑏 ∈ 𝑎 (𝑛‘𝑏) ≠ 𝑋 ↔ ∩ {𝑎 ∈ On ∣ ∀𝑛 ∈ 𝐹 ∃𝑏 ∈ 𝑎 (𝑛‘𝑏) ≠ 𝑋} ∈ On)
38	36, 37	sylib 207	. 2 ⊢ ((𝐹 ⊆ No ∧ 𝐹 ∈ 𝐴) → ∩ {𝑎 ∈ On ∣ ∀𝑛 ∈ 𝐹 ∃𝑏 ∈ 𝑎 (𝑛‘𝑏) ≠ 𝑋} ∈ On)
39	1, 38	syl5eqel 2692	1 ⊢ ((𝐹 ⊆ No ∧ 𝐹 ∈ 𝐴) → 𝐶 ∈ On)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900   ⊆ wss 3540  ∅c0 3874  {cpr 4127  ∪ cuni 4372  ∩ cint 4410  dom cdm 5038  ran crn 5039   “ cima 5041  Ord word 5639  Oncon0 5640  suc csuc 5642  Fun wfun 5798  ‘cfv 5804  1_𝑜c1o 7440  2_𝑜c2o 7441   No csur 31037   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-bday 31042

This theorem is referenced by:  nobndlem3  31093  nobndup  31099  nobnddown  31100