Theorem fin23lem24 9738

Description: Lemma for fin23 9805. 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 765	. . . . . 6 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → Ord 𝐴)
2		simplr 767	. . . . . . 7 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → 𝐵 ⊆ 𝐴)
3		simprl 769	. . . . . . 7 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → 𝐶 ∈ 𝐵)
4	2, 3	sseldd 3967	. . . . . 6 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → 𝐶 ∈ 𝐴)
5		ordelord 6207	. . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝐶 ∈ 𝐴) → Ord 𝐶)
6	1, 4, 5	syl2anc 586	. . . . 5 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → Ord 𝐶)
7		simprr 771	. . . . . . 7 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → 𝐷 ∈ 𝐵)
8	2, 7	sseldd 3967	. . . . . 6 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → 𝐷 ∈ 𝐴)
9		ordelord 6207	. . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝐷 ∈ 𝐴) → Ord 𝐷)
10	1, 8, 9	syl2anc 586	. . . . 5 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → Ord 𝐷)
11		ordtri3 6221	. . . . . 6 ⊢ ((Ord 𝐶 ∧ Ord 𝐷) → (𝐶 = 𝐷 ↔ ¬ (𝐶 ∈ 𝐷 ∨ 𝐷 ∈ 𝐶)))
12	11	necon2abid 3058	. . . . 5 ⊢ ((Ord 𝐶 ∧ Ord 𝐷) → ((𝐶 ∈ 𝐷 ∨ 𝐷 ∈ 𝐶) ↔ 𝐶 ≠ 𝐷))
13	6, 10, 12	syl2anc 586	. . . 4 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → ((𝐶 ∈ 𝐷 ∨ 𝐷 ∈ 𝐶) ↔ 𝐶 ≠ 𝐷))
14		simpr 487	. . . . . . . . 9 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐶 ∈ 𝐷) → 𝐶 ∈ 𝐷)
15		simplrl 775	. . . . . . . . 9 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐶 ∈ 𝐷) → 𝐶 ∈ 𝐵)
16	14, 15	elind 4170	. . . . . . . 8 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐶 ∈ 𝐷) → 𝐶 ∈ (𝐷 ∩ 𝐵))
17	6	adantr 483	. . . . . . . . . 10 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐶 ∈ 𝐷) → Ord 𝐶)
18		ordirr 6203	. . . . . . . . . 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 4171	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝐶 ∩ 𝐵) → 𝐶 ∈ 𝐶)
21	19, 20	nsyl 142	. . . . . . . 8 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐶 ∈ 𝐷) → ¬ 𝐶 ∈ (𝐶 ∩ 𝐵))
22		nelne1 3113	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝐷 ∩ 𝐵) ∧ ¬ 𝐶 ∈ (𝐶 ∩ 𝐵)) → (𝐷 ∩ 𝐵) ≠ (𝐶 ∩ 𝐵))
23	16, 21, 22	syl2anc 586	. . . . . . 7 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐶 ∈ 𝐷) → (𝐷 ∩ 𝐵) ≠ (𝐶 ∩ 𝐵))
24	23	necomd 3071	. . . . . 6 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐶 ∈ 𝐷) → (𝐶 ∩ 𝐵) ≠ (𝐷 ∩ 𝐵))
25		simpr 487	. . . . . . . 8 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐷 ∈ 𝐶) → 𝐷 ∈ 𝐶)
26		simplrr 776	. . . . . . . 8 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐷 ∈ 𝐶) → 𝐷 ∈ 𝐵)
27	25, 26	elind 4170	. . . . . . 7 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐷 ∈ 𝐶) → 𝐷 ∈ (𝐶 ∩ 𝐵))
28	10	adantr 483	. . . . . . . . 9 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐷 ∈ 𝐶) → Ord 𝐷)
29		ordirr 6203	. . . . . . . . 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 4171	. . . . . . . 8 ⊢ (𝐷 ∈ (𝐷 ∩ 𝐵) → 𝐷 ∈ 𝐷)
32	30, 31	nsyl 142	. . . . . . 7 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐷 ∈ 𝐶) → ¬ 𝐷 ∈ (𝐷 ∩ 𝐵))
33		nelne1 3113	. . . . . . 7 ⊢ ((𝐷 ∈ (𝐶 ∩ 𝐵) ∧ ¬ 𝐷 ∈ (𝐷 ∩ 𝐵)) → (𝐶 ∩ 𝐵) ≠ (𝐷 ∩ 𝐵))
34	27, 32, 33	syl2anc 586	. . . . . 6 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ 𝐷 ∈ 𝐶) → (𝐶 ∩ 𝐵) ≠ (𝐷 ∩ 𝐵))
35	24, 34	jaodan 954	. . . . 5 ⊢ ((((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) ∧ (𝐶 ∈ 𝐷 ∨ 𝐷 ∈ 𝐶)) → (𝐶 ∩ 𝐵) ≠ (𝐷 ∩ 𝐵))
36	35	ex 415	. . . 4 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → ((𝐶 ∈ 𝐷 ∨ 𝐷 ∈ 𝐶) → (𝐶 ∩ 𝐵) ≠ (𝐷 ∩ 𝐵)))
37	13, 36	sylbird 262	. . 3 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → (𝐶 ≠ 𝐷 → (𝐶 ∩ 𝐵) ≠ (𝐷 ∩ 𝐵)))
38	37	necon4d 3040	. 2 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → ((𝐶 ∩ 𝐵) = (𝐷 ∩ 𝐵) → 𝐶 = 𝐷))
39		ineq1 4180	. 2 ⊢ (𝐶 = 𝐷 → (𝐶 ∩ 𝐵) = (𝐷 ∩ 𝐵))
40	38, 39	impbid1 227	1 ⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → ((𝐶 ∩ 𝐵) = (𝐷 ∩ 𝐵) ↔ 𝐶 = 𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1533   ∈ wcel 2110   ≠ wne 3016   ∩ cin 3934   ⊆ wss 3935  Ord word 6184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-tr 5165  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-ord 6188

This theorem is referenced by:  fin23lem23  9742