Theorem nobndlem5 31095

Description: Lemma for nobndup 31099 and nobnddown 31100. There is always a minimal value of a surreal that is not a given sign. (Contributed by Scott Fenton, 3-Aug-2011.)

Hypothesis

Ref	Expression
nobndlem4.1	⊢ 𝑋 ∈ {1𝑜, 2𝑜}

Assertion

Ref	Expression
nobndlem5	⊢ (𝐴 ∈ No → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋}) ≠ 𝑋)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑋

Proof of Theorem nobndlem5

Step	Hyp	Ref	Expression
1		ssrab2 3650	. . . 4 ⊢ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ⊆ On
2		nobndlem4.1	. . . . . . . 8 ⊢ 𝑋 ∈ {1𝑜, 2𝑜}
3	2	nosgnn0i 31056	. . . . . . 7 ⊢ ∅ ≠ 𝑋
4		fvnobday 31081	. . . . . . . 8 ⊢ (𝐴 ∈ No → (𝐴‘( bday ‘𝐴)) = ∅)
5	4	neeq1d 2841	. . . . . . 7 ⊢ (𝐴 ∈ No → ((𝐴‘( bday ‘𝐴)) ≠ 𝑋 ↔ ∅ ≠ 𝑋))
6	3, 5	mpbiri 247	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴‘( bday ‘𝐴)) ≠ 𝑋)
7		bdayelon 31079	. . . . . . 7 ⊢ ( bday ‘𝐴) ∈ On
8		fveq2 6103	. . . . . . . . 9 ⊢ (𝑥 = ( bday ‘𝐴) → (𝐴‘𝑥) = (𝐴‘( bday ‘𝐴)))
9	8	neeq1d 2841	. . . . . . . 8 ⊢ (𝑥 = ( bday ‘𝐴) → ((𝐴‘𝑥) ≠ 𝑋 ↔ (𝐴‘( bday ‘𝐴)) ≠ 𝑋))
10	9	elrab3 3332	. . . . . . 7 ⊢ (( bday ‘𝐴) ∈ On → (( bday ‘𝐴) ∈ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ↔ (𝐴‘( bday ‘𝐴)) ≠ 𝑋))
11	7, 10	ax-mp 5	. . . . . 6 ⊢ (( bday ‘𝐴) ∈ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ↔ (𝐴‘( bday ‘𝐴)) ≠ 𝑋)
12	6, 11	sylibr 223	. . . . 5 ⊢ (𝐴 ∈ No → ( bday ‘𝐴) ∈ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋})
13		ne0i 3880	. . . . 5 ⊢ (( bday ‘𝐴) ∈ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} → {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ≠ ∅)
14	12, 13	syl 17	. . . 4 ⊢ (𝐴 ∈ No → {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ≠ ∅)
15		onint 6887	. . . 4 ⊢ (({𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ⊆ On ∧ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ≠ ∅) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ∈ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋})
16	1, 14, 15	sylancr 694	. . 3 ⊢ (𝐴 ∈ No → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ∈ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋})
17		nfrab1 3099	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋}
18	17	nfint 4421	. . . 4 ⊢ Ⅎ𝑥∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋}
19		nfcv 2751	. . . 4 ⊢ Ⅎ𝑥On
20		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑥𝐴
21	20, 18	nffv 6110	. . . . 5 ⊢ Ⅎ𝑥(𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋})
22		nfcv 2751	. . . . 5 ⊢ Ⅎ𝑥𝑋
23	21, 22	nfne 2882	. . . 4 ⊢ Ⅎ𝑥(𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋}) ≠ 𝑋
24		fveq2 6103	. . . . 5 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} → (𝐴‘𝑥) = (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋}))
25	24	neeq1d 2841	. . . 4 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} → ((𝐴‘𝑥) ≠ 𝑋 ↔ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋}) ≠ 𝑋))
26	18, 19, 23, 25	elrabf 3329	. . 3 ⊢ (∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ∈ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ↔ (∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ∈ On ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋}) ≠ 𝑋))
27	16, 26	sylib 207	. 2 ⊢ (𝐴 ∈ No → (∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ∈ On ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋}) ≠ 𝑋))
28	27	simprd 478	1 ⊢ (𝐴 ∈ No → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋}) ≠ 𝑋)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {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:  nobndup  31099  nobnddown  31100