Theorem fin23lem24 10391

Description: Lemma for fin23 10458. 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 766	. . . . . 6 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → Ord 𝐴)
2		simplr 768	. . . . . . 7 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → 𝐵 ⊆ 𝐴)
3		simprl 770	. . . . . . 7 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → 𝐶 ∈ 𝐵)
4	2, 3	sseldd 4009	. . . . . 6 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → 𝐶 ∈ 𝐴)
5		ordelord 6417	. . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝐶 ∈ 𝐴) → Ord 𝐶)
6	1, 4, 5	syl2anc 583	. . . . 5 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → Ord 𝐶)
7		simprr 772	. . . . . . 7 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → 𝐷 ∈ 𝐵)
8	2, 7	sseldd 4009	. . . . . 6 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → 𝐷 ∈ 𝐴)
9		ordelord 6417	. . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝐷 ∈ 𝐴) → Ord 𝐷)
10	1, 8, 9	syl2anc 583	. . . . 5 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → Ord 𝐷)
11		ordtri3 6431	. . . . . 6 ⊢ ((Ord 𝐶 ∧ Ord 𝐷) → (𝐶 = 𝐷 ↔ ¬ (𝐶 ∈ 𝐷 ∨ 𝐷 ∈ 𝐶)))
12	11	necon2abid 2989	. . . . 5 ⊢ ((Ord 𝐶 ∧ Ord 𝐷) → ((𝐶 ∈ 𝐷 ∨ 𝐷 ∈ 𝐶) ↔ 𝐶 ≠ 𝐷))
13	6, 10, 12	syl2anc 583	. . . 4 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → ((𝐶 ∈ 𝐷 ∨ 𝐷 ∈ 𝐶) ↔ 𝐶 ≠ 𝐷))
14		simpr 484	. . . . . . . . 9 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐶 ∈ 𝐷) → 𝐶 ∈ 𝐷)
15		simplrl 776	. . . . . . . . 9 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐶 ∈ 𝐷) → 𝐶 ∈ 𝐵)
16	14, 15	elind 4223	. . . . . . . 8 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐶 ∈ 𝐷) → 𝐶 ∈ (𝐷 ∩ 𝐵))
17	6	adantr 480	. . . . . . . . . 10 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐶 ∈ 𝐷) → Ord 𝐶)
18		ordirr 6413	. . . . . . . . . 10 ⊢ (Ord 𝐶 → ¬ 𝐶 ∈ 𝐶)
19	17, 18	syl 17	. . . . . . . . 9 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐶 ∈ 𝐷) → ¬ 𝐶 ∈ 𝐶)
20		elinel1 4224	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝐶 ∩ 𝐵) → 𝐶 ∈ 𝐶)
21	19, 20	nsyl 140	. . . . . . . 8 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐶 ∈ 𝐷) → ¬ 𝐶 ∈ (𝐶 ∩ 𝐵))
22		nelne1 3045	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝐷 ∩ 𝐵) ∧ ¬ 𝐶 ∈ (𝐶 ∩ 𝐵)) → (𝐷 ∩ 𝐵) ≠ (𝐶 ∩ 𝐵))
23	16, 21, 22	syl2anc 583	. . . . . . 7 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐶 ∈ 𝐷) → (𝐷 ∩ 𝐵) ≠ (𝐶 ∩ 𝐵))
24	23	necomd 3002	. . . . . 6 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐶 ∈ 𝐷) → (𝐶 ∩ 𝐵) ≠ (𝐷 ∩ 𝐵))
25		simpr 484	. . . . . . . 8 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐷 ∈ 𝐶) → 𝐷 ∈ 𝐶)
26		simplrr 777	. . . . . . . 8 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐷 ∈ 𝐶) → 𝐷 ∈ 𝐵)
27	25, 26	elind 4223	. . . . . . 7 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐷 ∈ 𝐶) → 𝐷 ∈ (𝐶 ∩ 𝐵))
28	10	adantr 480	. . . . . . . . 9 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐷 ∈ 𝐶) → Ord 𝐷)
29		ordirr 6413	. . . . . . . . 9 ⊢ (Ord 𝐷 → ¬ 𝐷 ∈ 𝐷)
30	28, 29	syl 17	. . . . . . . 8 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐷 ∈ 𝐶) → ¬ 𝐷 ∈ 𝐷)
31		elinel1 4224	. . . . . . . 8 ⊢ (𝐷 ∈ (𝐷 ∩ 𝐵) → 𝐷 ∈ 𝐷)
32	30, 31	nsyl 140	. . . . . . 7 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐷 ∈ 𝐶) → ¬ 𝐷 ∈ (𝐷 ∩ 𝐵))
33		nelne1 3045	. . . . . . 7 ⊢ ((𝐷 ∈ (𝐶 ∩ 𝐵) ∧ ¬ 𝐷 ∈ (𝐷 ∩ 𝐵)) → (𝐶 ∩ 𝐵) ≠ (𝐷 ∩ 𝐵))
34	27, 32, 33	syl2anc 583	. . . . . 6 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐷 ∈ 𝐶) → (𝐶 ∩ 𝐵) ≠ (𝐷 ∩ 𝐵))
35	24, 34	jaodan 958	. . . . 5 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ (𝐶 ∈ 𝐷 ∨ 𝐷 ∈ 𝐶)) → (𝐶 ∩ 𝐵) ≠ (𝐷 ∩ 𝐵))
36	35	ex 412	. . . 4 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → ((𝐶 ∈ 𝐷 ∨ 𝐷 ∈ 𝐶) → (𝐶 ∩ 𝐵) ≠ (𝐷 ∩ 𝐵)))
37	13, 36	sylbird 260	. . 3 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → (𝐶 ≠ 𝐷 → (𝐶 ∩ 𝐵) ≠ (𝐷 ∩ 𝐵)))
38	37	necon4d 2970	. 2 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → ((𝐶 ∩ 𝐵) = (𝐷 ∩ 𝐵) → 𝐶 = 𝐷))
39		ineq1 4234	. 2 ⊢ (𝐶 = 𝐷 → (𝐶 ∩ 𝐵) = (𝐷 ∩ 𝐵))
40	38, 39	impbid1 225	1 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → ((𝐶 ∩ 𝐵) = (𝐷 ∩ 𝐵) ↔ 𝐶 = 𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846   = wceq 1537   ∈ wcel 2108   ≠ wne 2946   ∩ cin 3975   ⊆ wss 3976  Ord word 6394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-ord 6398

This theorem is referenced by:  fin23lem23  10395