Theorem nodenselem5 27188

Description: Lemma for nodense 27192. If the birthdays of two distinct surreals are equal, then the ordinal from nodenselem4 27187 is an element of that birthday. (Contributed by Scott Fenton, 16-Jun-2011.)

Assertion

Ref	Expression
nodenselem5	⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (( bday ‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐴))

Distinct variable groups:   𝐴,𝑎   𝐵,𝑎

Proof of Theorem nodenselem5

Step	Hyp	Ref	Expression
1		simpll 765	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (( bday ‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → 𝐴 ∈ No )
2		simplr 767	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (( bday ‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → 𝐵 ∈ No )
3		sltso 27176	. . . . . . . . . 10 ⊢ <s Or No
4		sonr 5611	. . . . . . . . . 10 ⊢ (( <s Or No ∧ 𝐴 ∈ No ) → ¬ 𝐴 <s 𝐴)
5	3, 4	mpan 688	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ¬ 𝐴 <s 𝐴)
6		breq2 5152	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → (𝐴 <s 𝐴 ↔ 𝐴 <s 𝐵))
7	6	notbid 317	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (¬ 𝐴 <s 𝐴 ↔ ¬ 𝐴 <s 𝐵))
8	5, 7	syl5ibcom 244	. . . . . . . 8 ⊢ (𝐴 ∈ No → (𝐴 = 𝐵 → ¬ 𝐴 <s 𝐵))
9	8	necon2ad 2955	. . . . . . 7 ⊢ (𝐴 ∈ No → (𝐴 <s 𝐵 → 𝐴 ≠ 𝐵))
10	9	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 → 𝐴 ≠ 𝐵))
11	10	imp 407	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 <s 𝐵) → 𝐴 ≠ 𝐵)
12	11	adantrl 714	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (( bday ‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → 𝐴 ≠ 𝐵)
13		nosepdm 27184	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ (dom 𝐴 ∪ dom 𝐵))
14	1, 2, 12, 13	syl3anc 1371	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (( bday ‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ (dom 𝐴 ∪ dom 𝐵))
15		simprl 769	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (( bday ‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ( bday ‘𝐴) = ( bday ‘𝐵))
16	15	uneq2d 4163	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (( bday ‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → (( bday ‘𝐴) ∪ ( bday ‘𝐴)) = (( bday ‘𝐴) ∪ ( bday ‘𝐵)))
17		unidm 4152	. . . . 5 ⊢ (( bday ‘𝐴) ∪ ( bday ‘𝐴)) = ( bday ‘𝐴)
18	16, 17	eqtr3di 2787	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (( bday ‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → (( bday ‘𝐴) ∪ ( bday ‘𝐵)) = ( bday ‘𝐴))
19		bdayval 27148	. . . . . 6 ⊢ (𝐴 ∈ No → ( bday ‘𝐴) = dom 𝐴)
20	1, 19	syl 17	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (( bday ‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ( bday ‘𝐴) = dom 𝐴)
21		bdayval 27148	. . . . . 6 ⊢ (𝐵 ∈ No → ( bday ‘𝐵) = dom 𝐵)
22	2, 21	syl 17	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (( bday ‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ( bday ‘𝐵) = dom 𝐵)
23	20, 22	uneq12d 4164	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (( bday ‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → (( bday ‘𝐴) ∪ ( bday ‘𝐵)) = (dom 𝐴 ∪ dom 𝐵))
24	18, 23, 20	3eqtr3d 2780	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (( bday ‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → (dom 𝐴 ∪ dom 𝐵) = dom 𝐴)
25	14, 24	eleqtrd 2835	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (( bday ‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ dom 𝐴)
26	25, 20	eleqtrrd 2836	1 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (( bday ‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵)) → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  {crab 3432   ∪ cun 3946  ∩ cint 4950   class class class wbr 5148   Or wor 5587  dom cdm 5676  Oncon0 6364  ‘cfv 6543   No csur 27140   <s cslt 27141   bday cbday 27142

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-1o 8465  df-2o 8466  df-no 27143  df-slt 27144  df-bday 27145

This theorem is referenced by:  nodenselem8  27191  nodense  27192