Theorem nodenselem4 31083

Description: Lemma for nodense 31088. Show that a particular abstraction is an ordinal. (Contributed by Scott Fenton, 16-Jun-2011.)

Assertion

Ref	Expression
nodenselem4	⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 <s 𝐵) → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ On)

Distinct variable groups:   𝐴,𝑎   𝐵,𝑎

Proof of Theorem nodenselem4

Step	Hyp	Ref	Expression
1		ssrab2 3650	. 2 ⊢ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ⊆ On
2		sltirr 31069	. . . . . . 7 ⊢ (𝐴 ∈ No → ¬ 𝐴 <s 𝐴)
3		breq2 4587	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (𝐴 <s 𝐴 ↔ 𝐴 <s 𝐵))
4	3	biimprcd 239	. . . . . . . 8 ⊢ (𝐴 <s 𝐵 → (𝐴 = 𝐵 → 𝐴 <s 𝐴))
5	4	con3d 147	. . . . . . 7 ⊢ (𝐴 <s 𝐵 → (¬ 𝐴 <s 𝐴 → ¬ 𝐴 = 𝐵))
6	2, 5	syl5com 31	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴 <s 𝐵 → ¬ 𝐴 = 𝐵))
7	6	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 → ¬ 𝐴 = 𝐵))
8		nofnbday 31049	. . . . . . . 8 ⊢ (𝐴 ∈ No → 𝐴 Fn ( bday ‘𝐴))
9		nofnbday 31049	. . . . . . . 8 ⊢ (𝐵 ∈ No → 𝐵 Fn ( bday ‘𝐵))
10		eqfnfv2 6220	. . . . . . . 8 ⊢ ((𝐴 Fn ( bday ‘𝐴) ∧ 𝐵 Fn ( bday ‘𝐵)) → (𝐴 = 𝐵 ↔ (( bday ‘𝐴) = ( bday ‘𝐵) ∧ ∀𝑎 ∈ ( bday ‘𝐴)(𝐴‘𝑎) = (𝐵‘𝑎))))
11	8, 9, 10	syl2an 493	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 = 𝐵 ↔ (( bday ‘𝐴) = ( bday ‘𝐵) ∧ ∀𝑎 ∈ ( bday ‘𝐴)(𝐴‘𝑎) = (𝐵‘𝑎))))
12	11	notbid 307	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (¬ 𝐴 = 𝐵 ↔ ¬ (( bday ‘𝐴) = ( bday ‘𝐵) ∧ ∀𝑎 ∈ ( bday ‘𝐴)(𝐴‘𝑎) = (𝐵‘𝑎))))
13		ianor 508	. . . . . . 7 ⊢ (¬ (( bday ‘𝐴) = ( bday ‘𝐵) ∧ ∀𝑎 ∈ ( bday ‘𝐴)(𝐴‘𝑎) = (𝐵‘𝑎)) ↔ (¬ ( bday ‘𝐴) = ( bday ‘𝐵) ∨ ¬ ∀𝑎 ∈ ( bday ‘𝐴)(𝐴‘𝑎) = (𝐵‘𝑎)))
14		bdayelon 31079	. . . . . . . . . . . 12 ⊢ ( bday ‘𝐴) ∈ On
15	14	onordi 5749	. . . . . . . . . . 11 ⊢ Ord ( bday ‘𝐴)
16		bdayelon 31079	. . . . . . . . . . . 12 ⊢ ( bday ‘𝐵) ∈ On
17	16	onordi 5749	. . . . . . . . . . 11 ⊢ Ord ( bday ‘𝐵)
18		ordtri3 5676	. . . . . . . . . . 11 ⊢ ((Ord ( bday ‘𝐴) ∧ Ord ( bday ‘𝐵)) → (( bday ‘𝐴) = ( bday ‘𝐵) ↔ ¬ (( bday ‘𝐴) ∈ ( bday ‘𝐵) ∨ ( bday ‘𝐵) ∈ ( bday ‘𝐴))))
19	15, 17, 18	mp2an 704	. . . . . . . . . 10 ⊢ (( bday ‘𝐴) = ( bday ‘𝐵) ↔ ¬ (( bday ‘𝐴) ∈ ( bday ‘𝐵) ∨ ( bday ‘𝐵) ∈ ( bday ‘𝐴)))
20	19	con2bii 346	. . . . . . . . 9 ⊢ ((( bday ‘𝐴) ∈ ( bday ‘𝐵) ∨ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) ↔ ¬ ( bday ‘𝐴) = ( bday ‘𝐵))
21		nodenselem3 31082	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( bday ‘𝐴) ∈ ( bday ‘𝐵) → ∃𝑎 ∈ On (𝐴‘𝑎) ≠ (𝐵‘𝑎)))
22		nodenselem3 31082	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ) → (( bday ‘𝐵) ∈ ( bday ‘𝐴) → ∃𝑎 ∈ On (𝐵‘𝑎) ≠ (𝐴‘𝑎)))
23		necom 2835	. . . . . . . . . . . . 13 ⊢ ((𝐵‘𝑎) ≠ (𝐴‘𝑎) ↔ (𝐴‘𝑎) ≠ (𝐵‘𝑎))
24	23	rexbii 3023	. . . . . . . . . . . 12 ⊢ (∃𝑎 ∈ On (𝐵‘𝑎) ≠ (𝐴‘𝑎) ↔ ∃𝑎 ∈ On (𝐴‘𝑎) ≠ (𝐵‘𝑎))
25	22, 24	syl6ib 240	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ) → (( bday ‘𝐵) ∈ ( bday ‘𝐴) → ∃𝑎 ∈ On (𝐴‘𝑎) ≠ (𝐵‘𝑎)))
26	25	ancoms 468	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( bday ‘𝐵) ∈ ( bday ‘𝐴) → ∃𝑎 ∈ On (𝐴‘𝑎) ≠ (𝐵‘𝑎)))
27	21, 26	jaod 394	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((( bday ‘𝐴) ∈ ( bday ‘𝐵) ∨ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → ∃𝑎 ∈ On (𝐴‘𝑎) ≠ (𝐵‘𝑎)))
28	20, 27	syl5bir 232	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (¬ ( bday ‘𝐴) = ( bday ‘𝐵) → ∃𝑎 ∈ On (𝐴‘𝑎) ≠ (𝐵‘𝑎)))
29		rexnal 2978	. . . . . . . . . 10 ⊢ (∃𝑎 ∈ ( bday ‘𝐴) ¬ (𝐴‘𝑎) = (𝐵‘𝑎) ↔ ¬ ∀𝑎 ∈ ( bday ‘𝐴)(𝐴‘𝑎) = (𝐵‘𝑎))
30	14	onssi 6929	. . . . . . . . . . . 12 ⊢ ( bday ‘𝐴) ⊆ On
31		ssrexv 3630	. . . . . . . . . . . 12 ⊢ (( bday ‘𝐴) ⊆ On → (∃𝑎 ∈ ( bday ‘𝐴) ¬ (𝐴‘𝑎) = (𝐵‘𝑎) → ∃𝑎 ∈ On ¬ (𝐴‘𝑎) = (𝐵‘𝑎)))
32	30, 31	ax-mp 5	. . . . . . . . . . 11 ⊢ (∃𝑎 ∈ ( bday ‘𝐴) ¬ (𝐴‘𝑎) = (𝐵‘𝑎) → ∃𝑎 ∈ On ¬ (𝐴‘𝑎) = (𝐵‘𝑎))
33		df-ne 2782	. . . . . . . . . . . 12 ⊢ ((𝐴‘𝑎) ≠ (𝐵‘𝑎) ↔ ¬ (𝐴‘𝑎) = (𝐵‘𝑎))
34	33	rexbii 3023	. . . . . . . . . . 11 ⊢ (∃𝑎 ∈ On (𝐴‘𝑎) ≠ (𝐵‘𝑎) ↔ ∃𝑎 ∈ On ¬ (𝐴‘𝑎) = (𝐵‘𝑎))
35	32, 34	sylibr 223	. . . . . . . . . 10 ⊢ (∃𝑎 ∈ ( bday ‘𝐴) ¬ (𝐴‘𝑎) = (𝐵‘𝑎) → ∃𝑎 ∈ On (𝐴‘𝑎) ≠ (𝐵‘𝑎))
36	29, 35	sylbir 224	. . . . . . . . 9 ⊢ (¬ ∀𝑎 ∈ ( bday ‘𝐴)(𝐴‘𝑎) = (𝐵‘𝑎) → ∃𝑎 ∈ On (𝐴‘𝑎) ≠ (𝐵‘𝑎))
37	36	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (¬ ∀𝑎 ∈ ( bday ‘𝐴)(𝐴‘𝑎) = (𝐵‘𝑎) → ∃𝑎 ∈ On (𝐴‘𝑎) ≠ (𝐵‘𝑎)))
38	28, 37	jaod 394	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((¬ ( bday ‘𝐴) = ( bday ‘𝐵) ∨ ¬ ∀𝑎 ∈ ( bday ‘𝐴)(𝐴‘𝑎) = (𝐵‘𝑎)) → ∃𝑎 ∈ On (𝐴‘𝑎) ≠ (𝐵‘𝑎)))
39	13, 38	syl5bi 231	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (¬ (( bday ‘𝐴) = ( bday ‘𝐵) ∧ ∀𝑎 ∈ ( bday ‘𝐴)(𝐴‘𝑎) = (𝐵‘𝑎)) → ∃𝑎 ∈ On (𝐴‘𝑎) ≠ (𝐵‘𝑎)))
40	12, 39	sylbid 229	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (¬ 𝐴 = 𝐵 → ∃𝑎 ∈ On (𝐴‘𝑎) ≠ (𝐵‘𝑎)))
41	7, 40	syld 46	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 → ∃𝑎 ∈ On (𝐴‘𝑎) ≠ (𝐵‘𝑎)))
42	41	imp 444	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 <s 𝐵) → ∃𝑎 ∈ On (𝐴‘𝑎) ≠ (𝐵‘𝑎))
43		rabn0 3912	. . 3 ⊢ ({𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ≠ ∅ ↔ ∃𝑎 ∈ On (𝐴‘𝑎) ≠ (𝐵‘𝑎))
44	42, 43	sylibr 223	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 <s 𝐵) → {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ≠ ∅)
45		oninton 6892	. 2 ⊢ (({𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ⊆ On ∧ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ≠ ∅) → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ On)
46	1, 44, 45	sylancr 694	1 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 <s 𝐵) → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ On)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900   ⊆ wss 3540  ∅c0 3874  ∩ cint 4410   class class class wbr 4583  Ord word 5639  Oncon0 5640   Fn wfn 5799  ‘cfv 5804   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:  nodenselem5  31084  nodenselem6  31085  nodenselem7  31086  nodense  31088