Theorem nobndlem6 31096

Description: Lemma for nobndup 31099 and nobnddown 31100. Given an element 𝐴 of 𝐹, then the first position where it differs from 𝑋 is strictly less than 𝐶. (Contributed by Scott Fenton, 3-Aug-2011.)

Hypotheses

Ref	Expression
nobndlem6.1	⊢ 𝑋 ∈ {1𝑜, 2𝑜}
nobndlem6.2	⊢ 𝐶 = ∩ {𝑎 ∈ On ∣ ∀𝑛 ∈ 𝐹 ∃𝑏 ∈ 𝑎 (𝑛‘𝑏) ≠ 𝑋}

Assertion

Ref	Expression
nobndlem6	⊢ ((𝐹 ⊆ No ∧ 𝐴 ∈ 𝐹) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ∈ 𝐶)

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

Allowed substitution hints:   𝐶(𝑥,𝑛,𝑎,𝑏)   𝐹(𝑥)

Proof of Theorem nobndlem6

Step	Hyp	Ref	Expression
1		bdayelon 31079	. . . . 5 ⊢ ( bday ‘𝐴) ∈ On
2		ssel2 3563	. . . . . 6 ⊢ ((𝐹 ⊆ No ∧ 𝐴 ∈ 𝐹) → 𝐴 ∈ No )
3		nobndlem6.1	. . . . . . . 8 ⊢ 𝑋 ∈ {1𝑜, 2𝑜}
4	3	nosgnn0i 31056	. . . . . . 7 ⊢ ∅ ≠ 𝑋
5		fvnobday 31081	. . . . . . . 8 ⊢ (𝐴 ∈ No → (𝐴‘( bday ‘𝐴)) = ∅)
6	5	neeq1d 2841	. . . . . . 7 ⊢ (𝐴 ∈ No → ((𝐴‘( bday ‘𝐴)) ≠ 𝑋 ↔ ∅ ≠ 𝑋))
7	4, 6	mpbiri 247	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴‘( bday ‘𝐴)) ≠ 𝑋)
8	2, 7	syl 17	. . . . 5 ⊢ ((𝐹 ⊆ No ∧ 𝐴 ∈ 𝐹) → (𝐴‘( bday ‘𝐴)) ≠ 𝑋)
9		fveq2 6103	. . . . . . 7 ⊢ (𝑥 = ( bday ‘𝐴) → (𝐴‘𝑥) = (𝐴‘( bday ‘𝐴)))
10	9	neeq1d 2841	. . . . . 6 ⊢ (𝑥 = ( bday ‘𝐴) → ((𝐴‘𝑥) ≠ 𝑋 ↔ (𝐴‘( bday ‘𝐴)) ≠ 𝑋))
11	10	rspcev 3282	. . . . 5 ⊢ ((( bday ‘𝐴) ∈ On ∧ (𝐴‘( bday ‘𝐴)) ≠ 𝑋) → ∃𝑥 ∈ On (𝐴‘𝑥) ≠ 𝑋)
12	1, 8, 11	sylancr 694	. . . 4 ⊢ ((𝐹 ⊆ No ∧ 𝐴 ∈ 𝐹) → ∃𝑥 ∈ On (𝐴‘𝑥) ≠ 𝑋)
13		onintrab2 6894	. . . 4 ⊢ (∃𝑥 ∈ On (𝐴‘𝑥) ≠ 𝑋 ↔ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ∈ On)
14	12, 13	sylib 207	. . 3 ⊢ ((𝐹 ⊆ No ∧ 𝐴 ∈ 𝐹) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ∈ On)
15		fveq1 6102	. . . . . . . . 9 ⊢ (𝑛 = 𝐴 → (𝑛‘𝑏) = (𝐴‘𝑏))
16	15	neeq1d 2841	. . . . . . . 8 ⊢ (𝑛 = 𝐴 → ((𝑛‘𝑏) ≠ 𝑋 ↔ (𝐴‘𝑏) ≠ 𝑋))
17	16	rexbidv 3034	. . . . . . 7 ⊢ (𝑛 = 𝐴 → (∃𝑏 ∈ 𝑎 (𝑛‘𝑏) ≠ 𝑋 ↔ ∃𝑏 ∈ 𝑎 (𝐴‘𝑏) ≠ 𝑋))
18	17	rspcv 3278	. . . . . 6 ⊢ (𝐴 ∈ 𝐹 → (∀𝑛 ∈ 𝐹 ∃𝑏 ∈ 𝑎 (𝑛‘𝑏) ≠ 𝑋 → ∃𝑏 ∈ 𝑎 (𝐴‘𝑏) ≠ 𝑋))
19	18	ad2antlr 759	. . . . 5 ⊢ (((𝐹 ⊆ No ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ On) → (∀𝑛 ∈ 𝐹 ∃𝑏 ∈ 𝑎 (𝑛‘𝑏) ≠ 𝑋 → ∃𝑏 ∈ 𝑎 (𝐴‘𝑏) ≠ 𝑋))
20	14	ad2antrr 758	. . . . . . 7 ⊢ ((((𝐹 ⊆ No ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ On) ∧ (𝑏 ∈ 𝑎 ∧ (𝐴‘𝑏) ≠ 𝑋)) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ∈ On)
21		simplr 788	. . . . . . 7 ⊢ ((((𝐹 ⊆ No ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ On) ∧ (𝑏 ∈ 𝑎 ∧ (𝐴‘𝑏) ≠ 𝑋)) → 𝑎 ∈ On)
22		onelon 5665	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ 𝑎) → 𝑏 ∈ On)
23	22	anim1i 590	. . . . . . . . . 10 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ 𝑎) ∧ (𝐴‘𝑏) ≠ 𝑋) → (𝑏 ∈ On ∧ (𝐴‘𝑏) ≠ 𝑋))
24	23	anasss 677	. . . . . . . . 9 ⊢ ((𝑎 ∈ On ∧ (𝑏 ∈ 𝑎 ∧ (𝐴‘𝑏) ≠ 𝑋)) → (𝑏 ∈ On ∧ (𝐴‘𝑏) ≠ 𝑋))
25		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑏 → (𝐴‘𝑥) = (𝐴‘𝑏))
26	25	neeq1d 2841	. . . . . . . . . 10 ⊢ (𝑥 = 𝑏 → ((𝐴‘𝑥) ≠ 𝑋 ↔ (𝐴‘𝑏) ≠ 𝑋))
27	26	intminss 4438	. . . . . . . . 9 ⊢ ((𝑏 ∈ On ∧ (𝐴‘𝑏) ≠ 𝑋) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ⊆ 𝑏)
28	24, 27	syl 17	. . . . . . . 8 ⊢ ((𝑎 ∈ On ∧ (𝑏 ∈ 𝑎 ∧ (𝐴‘𝑏) ≠ 𝑋)) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ⊆ 𝑏)
29	28	adantll 746	. . . . . . 7 ⊢ ((((𝐹 ⊆ No ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ On) ∧ (𝑏 ∈ 𝑎 ∧ (𝐴‘𝑏) ≠ 𝑋)) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ⊆ 𝑏)
30		simprl 790	. . . . . . 7 ⊢ ((((𝐹 ⊆ No ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ On) ∧ (𝑏 ∈ 𝑎 ∧ (𝐴‘𝑏) ≠ 𝑋)) → 𝑏 ∈ 𝑎)
31		ontr2 5689	. . . . . . . 8 ⊢ ((∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ∈ On ∧ 𝑎 ∈ On) → ((∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ⊆ 𝑏 ∧ 𝑏 ∈ 𝑎) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ∈ 𝑎))
32	31	imp 444	. . . . . . 7 ⊢ (((∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ∈ On ∧ 𝑎 ∈ On) ∧ (∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ⊆ 𝑏 ∧ 𝑏 ∈ 𝑎)) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ∈ 𝑎)
33	20, 21, 29, 30, 32	syl22anc 1319	. . . . . 6 ⊢ ((((𝐹 ⊆ No ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ On) ∧ (𝑏 ∈ 𝑎 ∧ (𝐴‘𝑏) ≠ 𝑋)) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ∈ 𝑎)
34	33	rexlimdvaa 3014	. . . . 5 ⊢ (((𝐹 ⊆ No ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ On) → (∃𝑏 ∈ 𝑎 (𝐴‘𝑏) ≠ 𝑋 → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ∈ 𝑎))
35	19, 34	syld 46	. . . 4 ⊢ (((𝐹 ⊆ No ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ On) → (∀𝑛 ∈ 𝐹 ∃𝑏 ∈ 𝑎 (𝑛‘𝑏) ≠ 𝑋 → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ∈ 𝑎))
36	35	ralrimiva 2949	. . 3 ⊢ ((𝐹 ⊆ No ∧ 𝐴 ∈ 𝐹) → ∀𝑎 ∈ On (∀𝑛 ∈ 𝐹 ∃𝑏 ∈ 𝑎 (𝑛‘𝑏) ≠ 𝑋 → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ∈ 𝑎))
37		elintrabg 4424	. . . 4 ⊢ (∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ∈ On → (∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ∈ ∩ {𝑎 ∈ On ∣ ∀𝑛 ∈ 𝐹 ∃𝑏 ∈ 𝑎 (𝑛‘𝑏) ≠ 𝑋} ↔ ∀𝑎 ∈ On (∀𝑛 ∈ 𝐹 ∃𝑏 ∈ 𝑎 (𝑛‘𝑏) ≠ 𝑋 → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ∈ 𝑎)))
38	37	biimpar 501	. . 3 ⊢ ((∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ∈ On ∧ ∀𝑎 ∈ On (∀𝑛 ∈ 𝐹 ∃𝑏 ∈ 𝑎 (𝑛‘𝑏) ≠ 𝑋 → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ∈ 𝑎)) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ∈ ∩ {𝑎 ∈ On ∣ ∀𝑛 ∈ 𝐹 ∃𝑏 ∈ 𝑎 (𝑛‘𝑏) ≠ 𝑋})
39	14, 36, 38	syl2anc 691	. 2 ⊢ ((𝐹 ⊆ No ∧ 𝐴 ∈ 𝐹) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ∈ ∩ {𝑎 ∈ On ∣ ∀𝑛 ∈ 𝐹 ∃𝑏 ∈ 𝑎 (𝑛‘𝑏) ≠ 𝑋})
40		nobndlem6.2	. 2 ⊢ 𝐶 = ∩ {𝑎 ∈ On ∣ ∀𝑛 ∈ 𝐹 ∃𝑏 ∈ 𝑎 (𝑛‘𝑏) ≠ 𝑋}
41	39, 40	syl6eleqr 2699	1 ⊢ ((𝐹 ⊆ No ∧ 𝐴 ∈ 𝐹) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900   ⊆ wss 3540  ∅c0 3874  {cpr 4127  ∩ cint 4410  Oncon0 5640  ‘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:  nobndlem7  31097  nobndup  31099  nobnddown  31100