Theorem nodenselem5 31084

Description: Lemma for nodense 31088. If the birthdays of two distinct surreals are equal, then the ordinal from nodenselem4 31083 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

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sltirr 31069	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ¬ 𝐴 <s 𝐴)
2		breq2 4587	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → (𝐴 <s 𝐴 ↔ 𝐴 <s 𝐵))
3	2	notbid 307	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (¬ 𝐴 <s 𝐴 ↔ ¬ 𝐴 <s 𝐵))
4	1, 3	syl5ibcom 234	. . . . . . . 8 ⊢ (𝐴 ∈ No → (𝐴 = 𝐵 → ¬ 𝐴 <s 𝐵))
5	4	con2d 128	. . . . . . 7 ⊢ (𝐴 ∈ No → (𝐴 <s 𝐵 → ¬ 𝐴 = 𝐵))
6	5	imp 444	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐴 <s 𝐵) → ¬ 𝐴 = 𝐵)
7	6	ad2ant2rl 781	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (dom 𝐴 = dom 𝐵 ∧ 𝐴 <s 𝐵)) → ¬ 𝐴 = 𝐵)
8		nofun 31046	. . . . . . . . 9 ⊢ (𝐴 ∈ No → Fun 𝐴)
9		nofun 31046	. . . . . . . . 9 ⊢ (𝐵 ∈ No → Fun 𝐵)
10		eqfunfv 6224	. . . . . . . . 9 ⊢ ((Fun 𝐴 ∧ Fun 𝐵) → (𝐴 = 𝐵 ↔ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥))))
11	8, 9, 10	syl2an 493	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 = 𝐵 ↔ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥))))
12	11	notbid 307	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (¬ 𝐴 = 𝐵 ↔ ¬ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥))))
13	12	adantr 480	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (dom 𝐴 = dom 𝐵 ∧ 𝐴 <s 𝐵)) → (¬ 𝐴 = 𝐵 ↔ ¬ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥))))
14		imnan 437	. . . . . . . . . . . 12 ⊢ ((dom 𝐴 = dom 𝐵 → ¬ ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥)) ↔ ¬ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥)))
15	14	biimpri 217	. . . . . . . . . . 11 ⊢ (¬ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥)) → (dom 𝐴 = dom 𝐵 → ¬ ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥)))
16	15	impcom 445	. . . . . . . . . 10 ⊢ ((dom 𝐴 = dom 𝐵 ∧ ¬ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥))) → ¬ ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥))
17		df-ne 2782	. . . . . . . . . . . . 13 ⊢ ((𝐴‘𝑥) ≠ (𝐵‘𝑥) ↔ ¬ (𝐴‘𝑥) = (𝐵‘𝑥))
18	17	rexbii 3023	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ dom 𝐴(𝐴‘𝑥) ≠ (𝐵‘𝑥) ↔ ∃𝑥 ∈ dom 𝐴 ¬ (𝐴‘𝑥) = (𝐵‘𝑥))
19		rexnal 2978	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ dom 𝐴 ¬ (𝐴‘𝑥) = (𝐵‘𝑥) ↔ ¬ ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥))
20	18, 19	bitri 263	. . . . . . . . . . 11 ⊢ (∃𝑥 ∈ dom 𝐴(𝐴‘𝑥) ≠ (𝐵‘𝑥) ↔ ¬ ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥))
21		nodenselem4 31083	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 <s 𝐵) → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ On)
22		eloni 5650	. . . . . . . . . . . . . . 15 ⊢ (∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ On → Ord ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})
23	21, 22	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 <s 𝐵) → Ord ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})
24	23	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 <s 𝐵) ∧ (𝑥 ∈ dom 𝐴 ∧ (𝐴‘𝑥) ≠ (𝐵‘𝑥))) → Ord ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)})
25		nodmord 31050	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ No → Ord dom 𝐴)
26	25	ad3antrrr 762	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 <s 𝐵) ∧ (𝑥 ∈ dom 𝐴 ∧ (𝐴‘𝑥) ≠ (𝐵‘𝑥))) → Ord dom 𝐴)
27		nodmon 31047	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
28		onelon 5665	. . . . . . . . . . . . . . . . . . 19 ⊢ ((dom 𝐴 ∈ On ∧ 𝑥 ∈ dom 𝐴) → 𝑥 ∈ On)
29	27, 28	sylan 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ No ∧ 𝑥 ∈ dom 𝐴) → 𝑥 ∈ On)
30	29	ex 449	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ No → (𝑥 ∈ dom 𝐴 → 𝑥 ∈ On))
31	30	ad2antrr 758	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 <s 𝐵) → (𝑥 ∈ dom 𝐴 → 𝑥 ∈ On))
32	31	anim1d 586	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 <s 𝐵) → ((𝑥 ∈ dom 𝐴 ∧ (𝐴‘𝑥) ≠ (𝐵‘𝑥)) → (𝑥 ∈ On ∧ (𝐴‘𝑥) ≠ (𝐵‘𝑥))))
33	32	imp 444	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 <s 𝐵) ∧ (𝑥 ∈ dom 𝐴 ∧ (𝐴‘𝑥) ≠ (𝐵‘𝑥))) → (𝑥 ∈ On ∧ (𝐴‘𝑥) ≠ (𝐵‘𝑥)))
34		fveq2 6103	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑥 → (𝐴‘𝑎) = (𝐴‘𝑥))
35		fveq2 6103	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑥 → (𝐵‘𝑎) = (𝐵‘𝑥))
36	34, 35	neeq12d 2843	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑥 → ((𝐴‘𝑎) ≠ (𝐵‘𝑎) ↔ (𝐴‘𝑥) ≠ (𝐵‘𝑥)))
37	36	intminss 4438	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ On ∧ (𝐴‘𝑥) ≠ (𝐵‘𝑥)) → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ⊆ 𝑥)
38	33, 37	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 <s 𝐵) ∧ (𝑥 ∈ dom 𝐴 ∧ (𝐴‘𝑥) ≠ (𝐵‘𝑥))) → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ⊆ 𝑥)
39		simprl 790	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 <s 𝐵) ∧ (𝑥 ∈ dom 𝐴 ∧ (𝐴‘𝑥) ≠ (𝐵‘𝑥))) → 𝑥 ∈ dom 𝐴)
40		ordtr2 5685	. . . . . . . . . . . . . 14 ⊢ ((Ord ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∧ Ord dom 𝐴) → ((∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝐴) → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ dom 𝐴))
41	40	imp 444	. . . . . . . . . . . . 13 ⊢ (((Ord ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∧ Ord dom 𝐴) ∧ (∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝐴)) → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ dom 𝐴)
42	24, 26, 38, 39, 41	syl22anc 1319	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 <s 𝐵) ∧ (𝑥 ∈ dom 𝐴 ∧ (𝐴‘𝑥) ≠ (𝐵‘𝑥))) → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ dom 𝐴)
43	42	rexlimdvaa 3014	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 <s 𝐵) → (∃𝑥 ∈ dom 𝐴(𝐴‘𝑥) ≠ (𝐵‘𝑥) → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ dom 𝐴))
44	20, 43	syl5bir 232	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 <s 𝐵) → (¬ ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥) → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ dom 𝐴))
45	16, 44	syl5 33	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 <s 𝐵) → ((dom 𝐴 = dom 𝐵 ∧ ¬ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥))) → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ dom 𝐴))
46	45	exp4b 630	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 → (dom 𝐴 = dom 𝐵 → (¬ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥)) → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ dom 𝐴))))
47	46	com23 84	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (dom 𝐴 = dom 𝐵 → (𝐴 <s 𝐵 → (¬ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥)) → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ dom 𝐴))))
48	47	imp32 448	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (dom 𝐴 = dom 𝐵 ∧ 𝐴 <s 𝐵)) → (¬ (dom 𝐴 = dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐴(𝐴‘𝑥) = (𝐵‘𝑥)) → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ dom 𝐴))
49	13, 48	sylbid 229	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (dom 𝐴 = dom 𝐵 ∧ 𝐴 <s 𝐵)) → (¬ 𝐴 = 𝐵 → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ dom 𝐴))
50	7, 49	mpd 15	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (dom 𝐴 = dom 𝐵 ∧ 𝐴 <s 𝐵)) → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ dom 𝐴)
51	50	ex 449	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((dom 𝐴 = dom 𝐵 ∧ 𝐴 <s 𝐵) → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ dom 𝐴))
52		bdayval 31045	. . . . 5 ⊢ (𝐴 ∈ No → ( bday ‘𝐴) = dom 𝐴)
53		bdayval 31045	. . . . 5 ⊢ (𝐵 ∈ No → ( bday ‘𝐵) = dom 𝐵)
54	52, 53	eqeqan12d 2626	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( bday ‘𝐴) = ( bday ‘𝐵) ↔ dom 𝐴 = dom 𝐵))
55	54	anbi1d 737	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((( bday ‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵) ↔ (dom 𝐴 = dom 𝐵 ∧ 𝐴 <s 𝐵)))
56	52	eleq2d 2673	. . . 4 ⊢ (𝐴 ∈ No → (∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐴) ↔ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ dom 𝐴))
57	56	adantr 480	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐴) ↔ ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ dom 𝐴))
58	51, 55, 57	3imtr4d 282	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((( bday ‘𝐴) = ( bday ‘𝐵) ∧ 𝐴 <s 𝐵) → ∩ {𝑎 ∈ On ∣ (𝐴‘𝑎) ≠ (𝐵‘𝑎)} ∈ ( bday ‘𝐴)))
59	58	imp 444	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   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900   ⊆ wss 3540  ∩ cint 4410   class class class wbr 4583  dom cdm 5038  Ord word 5639  Oncon0 5640  Fun wfun 5798  ‘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:  nodenselem6  31085  nodenselem8  31087  nodense  31088