Theorem fin23lem24 9144

Description: Lemma for fin23 9211. In a class of ordinals, each element is fully identified by those of its predecessors which also belong to the class. (Contributed by Stefan O'Rear, 1-Nov-2014.)

Assertion

Ref	Expression
fin23lem24	⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → ((𝐶 ∩ 𝐵) = (𝐷 ∩ 𝐵) ↔ 𝐶 = 𝐷))

Proof of Theorem fin23lem24

Step	Hyp	Ref	Expression
1		simpll 790	. . . . . 6 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → Ord 𝐴)
2		simplr 792	. . . . . . 7 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → 𝐵 ⊆ 𝐴)
3		simprl 794	. . . . . . 7 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → 𝐶 ∈ 𝐵)
4	2, 3	sseldd 3604	. . . . . 6 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → 𝐶 ∈ 𝐴)
5		ordelord 5745	. . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝐶 ∈ 𝐴) → Ord 𝐶)
6	1, 4, 5	syl2anc 693	. . . . 5 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → Ord 𝐶)
7		simprr 796	. . . . . . 7 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → 𝐷 ∈ 𝐵)
8	2, 7	sseldd 3604	. . . . . 6 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → 𝐷 ∈ 𝐴)
9		ordelord 5745	. . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝐷 ∈ 𝐴) → Ord 𝐷)
10	1, 8, 9	syl2anc 693	. . . . 5 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → Ord 𝐷)
11		ordtri3 5759	. . . . . 6 ⊢ ((Ord 𝐶 ∧ Ord 𝐷) → (𝐶 = 𝐷 ↔ ¬ (𝐶 ∈ 𝐷 ∨ 𝐷 ∈ 𝐶)))
12	11	necon2abid 2836	. . . . 5 ⊢ ((Ord 𝐶 ∧ Ord 𝐷) → ((𝐶 ∈ 𝐷 ∨ 𝐷 ∈ 𝐶) ↔ 𝐶 ≠ 𝐷))
13	6, 10, 12	syl2anc 693	. . . 4 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → ((𝐶 ∈ 𝐷 ∨ 𝐷 ∈ 𝐶) ↔ 𝐶 ≠ 𝐷))
14		simpr 477	. . . . . . . . 9 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐶 ∈ 𝐷) → 𝐶 ∈ 𝐷)
15		simplrl 800	. . . . . . . . 9 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐶 ∈ 𝐷) → 𝐶 ∈ 𝐵)
16	14, 15	elind 3798	. . . . . . . 8 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐶 ∈ 𝐷) → 𝐶 ∈ (𝐷 ∩ 𝐵))
17	6	adantr 481	. . . . . . . . . 10 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐶 ∈ 𝐷) → Ord 𝐶)
18		ordirr 5741	. . . . . . . . . 10 ⊢ (Ord 𝐶 → ¬ 𝐶 ∈ 𝐶)
19	17, 18	syl 17	. . . . . . . . 9 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐶 ∈ 𝐷) → ¬ 𝐶 ∈ 𝐶)
20		inss1 3833	. . . . . . . . . 10 ⊢ (𝐶 ∩ 𝐵) ⊆ 𝐶
21	20	sseli 3599	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝐶 ∩ 𝐵) → 𝐶 ∈ 𝐶)
22	19, 21	nsyl 135	. . . . . . . 8 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐶 ∈ 𝐷) → ¬ 𝐶 ∈ (𝐶 ∩ 𝐵))
23		nelne1 2890	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝐷 ∩ 𝐵) ∧ ¬ 𝐶 ∈ (𝐶 ∩ 𝐵)) → (𝐷 ∩ 𝐵) ≠ (𝐶 ∩ 𝐵))
24	16, 22, 23	syl2anc 693	. . . . . . 7 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐶 ∈ 𝐷) → (𝐷 ∩ 𝐵) ≠ (𝐶 ∩ 𝐵))
25	24	necomd 2849	. . . . . 6 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐶 ∈ 𝐷) → (𝐶 ∩ 𝐵) ≠ (𝐷 ∩ 𝐵))
26		simpr 477	. . . . . . . 8 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐷 ∈ 𝐶) → 𝐷 ∈ 𝐶)
27		simplrr 801	. . . . . . . 8 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐷 ∈ 𝐶) → 𝐷 ∈ 𝐵)
28	26, 27	elind 3798	. . . . . . 7 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐷 ∈ 𝐶) → 𝐷 ∈ (𝐶 ∩ 𝐵))
29	10	adantr 481	. . . . . . . . 9 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐷 ∈ 𝐶) → Ord 𝐷)
30		ordirr 5741	. . . . . . . . 9 ⊢ (Ord 𝐷 → ¬ 𝐷 ∈ 𝐷)
31	29, 30	syl 17	. . . . . . . 8 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐷 ∈ 𝐶) → ¬ 𝐷 ∈ 𝐷)
32		inss1 3833	. . . . . . . . 9 ⊢ (𝐷 ∩ 𝐵) ⊆ 𝐷
33	32	sseli 3599	. . . . . . . 8 ⊢ (𝐷 ∈ (𝐷 ∩ 𝐵) → 𝐷 ∈ 𝐷)
34	31, 33	nsyl 135	. . . . . . 7 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐷 ∈ 𝐶) → ¬ 𝐷 ∈ (𝐷 ∩ 𝐵))
35		nelne1 2890	. . . . . . 7 ⊢ ((𝐷 ∈ (𝐶 ∩ 𝐵) ∧ ¬ 𝐷 ∈ (𝐷 ∩ 𝐵)) → (𝐶 ∩ 𝐵) ≠ (𝐷 ∩ 𝐵))
36	28, 34, 35	syl2anc 693	. . . . . 6 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐷 ∈ 𝐶) → (𝐶 ∩ 𝐵) ≠ (𝐷 ∩ 𝐵))
37	25, 36	jaodan 826	. . . . 5 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ (𝐶 ∈ 𝐷 ∨ 𝐷 ∈ 𝐶)) → (𝐶 ∩ 𝐵) ≠ (𝐷 ∩ 𝐵))
38	37	ex 450	. . . 4 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → ((𝐶 ∈ 𝐷 ∨ 𝐷 ∈ 𝐶) → (𝐶 ∩ 𝐵) ≠ (𝐷 ∩ 𝐵)))
39	13, 38	sylbird 250	. . 3 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → (𝐶 ≠ 𝐷 → (𝐶 ∩ 𝐵) ≠ (𝐷 ∩ 𝐵)))
40	39	necon4d 2818	. 2 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → ((𝐶 ∩ 𝐵) = (𝐷 ∩ 𝐵) → 𝐶 = 𝐷))
41		ineq1 3807	. 2 ⊢ (𝐶 = 𝐷 → (𝐶 ∩ 𝐵) = (𝐷 ∩ 𝐵))
42	40, 41	impbid1 215	1 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → ((𝐶 ∩ 𝐵) = (𝐷 ∩ 𝐵) ↔ 𝐶 = 𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794   ∩ cin 3573   ⊆ wss 3574  Ord word 5722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726

This theorem is referenced by:  fin23lem23  9148