Theorem tz7.7 6388

Description: A transitive class belongs to an ordinal class iff it is strictly included in it. Proposition 7.7 of [TakeutiZaring] p. 37. (Contributed by NM, 5-May-1994.)

Assertion

Ref	Expression
tz7.7	⊢ ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵 ∈ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴)))

Proof of Theorem tz7.7

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordtr 6376	. . . 4 ⊢ (Ord 𝐴 → Tr 𝐴)
2		ordfr 6377	. . . 4 ⊢ (Ord 𝐴 → E Fr 𝐴)
3		tz7.2 5634	. . . . 5 ⊢ ((Tr 𝐴 ∧ E Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴))
4	3	3exp 1137	. . . 4 ⊢ (Tr 𝐴 → ( E Fr 𝐴 → (𝐵 ∈ 𝐴 → (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴))))
5	1, 2, 4	sylc 66	. . 3 ⊢ (Ord 𝐴 → (𝐵 ∈ 𝐴 → (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴)))
6	5	adantr 486	. 2 ⊢ ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵 ∈ 𝐴 → (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴)))
7		pssdifn0 4316	. . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴) → (𝐴 ∖ 𝐵) ≠ ∅)
8		difss 4083	. . . . . . . . . . . 12 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
9		tz7.5 6383	. . . . . . . . . . . 12 ⊢ ((Ord 𝐴 ∧ (𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)((𝐴 ∖ 𝐵) ∩ 𝑥) = ∅)
10	8, 9	mp3an2 1478	. . . . . . . . . . 11 ⊢ ((Ord 𝐴 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)((𝐴 ∖ 𝐵) ∩ 𝑥) = ∅)
11		eldifi 4078	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐴)
12		trss 5222	. . . . . . . . . . . . . . . . . 18 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
13		difin0ss 4321	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∖ 𝐵) ∩ 𝑥) = ∅ → (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵))
14	13	com12 33	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ⊆ 𝐴 → (((𝐴 ∖ 𝐵) ∩ 𝑥) = ∅ → 𝑥 ⊆ 𝐵))
15	11, 12, 14	syl56 37	. . . . . . . . . . . . . . . . 17 ⊢ (Tr 𝐴 → (𝑥 ∈ (𝐴 ∖ 𝐵) → (((𝐴 ∖ 𝐵) ∩ 𝑥) = ∅ → 𝑥 ⊆ 𝐵)))
16	1, 15	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (Ord 𝐴 → (𝑥 ∈ (𝐴 ∖ 𝐵) → (((𝐴 ∖ 𝐵) ∩ 𝑥) = ∅ → 𝑥 ⊆ 𝐵)))
17	16	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ (𝐴 ∖ 𝐵) → (((𝐴 ∖ 𝐵) ∩ 𝑥) = ∅ → 𝑥 ⊆ 𝐵)))
18	17	imp32 424	. . . . . . . . . . . . . 14 ⊢ ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ (𝐴 ∖ 𝐵) ∧ ((𝐴 ∖ 𝐵) ∩ 𝑥) = ∅)) → 𝑥 ⊆ 𝐵)
19		eleq1w 2844	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
20	19	biimpcd 252	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ 𝐵 → (𝑦 = 𝑥 → 𝑥 ∈ 𝐵))
21		eldifn 4079	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ¬ 𝑥 ∈ 𝐵)
22	20, 21	nsyli 158	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ 𝐵 → (𝑥 ∈ (𝐴 ∖ 𝐵) → ¬ 𝑦 = 𝑥))
23	22	imp 412	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) → ¬ 𝑦 = 𝑥)
24	23	adantll 727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) → ¬ 𝑦 = 𝑥)
25	24	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((Ord 𝐴 ∧ Tr 𝐵) ∧ ((𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵))) → ¬ 𝑦 = 𝑥)
26		trel 5220	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (Tr 𝐵 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵))
27	26	expcomd 422	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (Tr 𝐵 → (𝑦 ∈ 𝐵 → (𝑥 ∈ 𝑦 → 𝑥 ∈ 𝐵)))
28	27	imp 412	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((Tr 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ 𝑦 → 𝑥 ∈ 𝐵))
29	28, 21	nsyli 158	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((Tr 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ (𝐴 ∖ 𝐵) → ¬ 𝑥 ∈ 𝑦))
30	29	ex 418	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Tr 𝐵 → (𝑦 ∈ 𝐵 → (𝑥 ∈ (𝐴 ∖ 𝐵) → ¬ 𝑥 ∈ 𝑦)))
31	30	adantld 496	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Tr 𝐵 → ((𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ (𝐴 ∖ 𝐵) → ¬ 𝑥 ∈ 𝑦)))
32	31	imp32 424	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Tr 𝐵 ∧ ((𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵))) → ¬ 𝑥 ∈ 𝑦)
33	32	adantll 727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((Ord 𝐴 ∧ Tr 𝐵) ∧ ((𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵))) → ¬ 𝑥 ∈ 𝑦)
34		ordwe 6375	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Ord 𝐴 → E We 𝐴)
35		ssel2 3926	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐴)
36	35, 11	anim12i 625	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) → (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
37		wecmpep 5643	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (( E We 𝐴 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → (𝑦 ∈ 𝑥 ∨ 𝑦 = 𝑥 ∨ 𝑥 ∈ 𝑦))
38	34, 36, 37	syl2an 608	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Ord 𝐴 ∧ ((𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵))) → (𝑦 ∈ 𝑥 ∨ 𝑦 = 𝑥 ∨ 𝑥 ∈ 𝑦))
39	38	adantlr 728	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((Ord 𝐴 ∧ Tr 𝐵) ∧ ((𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵))) → (𝑦 ∈ 𝑥 ∨ 𝑦 = 𝑥 ∨ 𝑥 ∈ 𝑦))
40	25, 33, 39	ecase23d 1503	. . . . . . . . . . . . . . . . . . 19 ⊢ (((Ord 𝐴 ∧ Tr 𝐵) ∧ ((𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵))) → 𝑦 ∈ 𝑥)
41	40	exp44 443	. . . . . . . . . . . . . . . . . 18 ⊢ ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵 ⊆ 𝐴 → (𝑦 ∈ 𝐵 → (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑦 ∈ 𝑥))))
42	41	com34 92	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵 ⊆ 𝐴 → (𝑥 ∈ (𝐴 ∖ 𝐵) → (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝑥))))
43	42	imp31 423	. . . . . . . . . . . . . . . 16 ⊢ ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) → (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝑥))
44	43	ssrdv 3937	. . . . . . . . . . . . . . 15 ⊢ ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) → 𝐵 ⊆ 𝑥)
45	44	adantrr 730	. . . . . . . . . . . . . 14 ⊢ ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ (𝐴 ∖ 𝐵) ∧ ((𝐴 ∖ 𝐵) ∩ 𝑥) = ∅)) → 𝐵 ⊆ 𝑥)
46	18, 45	eqssd 3948	. . . . . . . . . . . . 13 ⊢ ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ (𝐴 ∖ 𝐵) ∧ ((𝐴 ∖ 𝐵) ∩ 𝑥) = ∅)) → 𝑥 = 𝐵)
47	11	ad2antrl 741	. . . . . . . . . . . . 13 ⊢ ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ (𝐴 ∖ 𝐵) ∧ ((𝐴 ∖ 𝐵) ∩ 𝑥) = ∅)) → 𝑥 ∈ 𝐴)
48	46, 47	eqeltrrd 2862	. . . . . . . . . . . 12 ⊢ ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ (𝐴 ∖ 𝐵) ∧ ((𝐴 ∖ 𝐵) ∩ 𝑥) = ∅)) → 𝐵 ∈ 𝐴)
49	48	rexlimdvaa 3165	. . . . . . . . . . 11 ⊢ (((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵 ⊆ 𝐴) → (∃𝑥 ∈ (𝐴 ∖ 𝐵)((𝐴 ∖ 𝐵) ∩ 𝑥) = ∅ → 𝐵 ∈ 𝐴))
50	10, 49	syl5 35	. . . . . . . . . 10 ⊢ (((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵 ⊆ 𝐴) → ((Ord 𝐴 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → 𝐵 ∈ 𝐴))
51	50	exp4b 436	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵 ⊆ 𝐴 → (Ord 𝐴 → ((𝐴 ∖ 𝐵) ≠ ∅ → 𝐵 ∈ 𝐴))))
52	51	com23 87	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ Tr 𝐵) → (Ord 𝐴 → (𝐵 ⊆ 𝐴 → ((𝐴 ∖ 𝐵) ≠ ∅ → 𝐵 ∈ 𝐴))))
53	52	adantrd 497	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ Tr 𝐵) → ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵 ⊆ 𝐴 → ((𝐴 ∖ 𝐵) ≠ ∅ → 𝐵 ∈ 𝐴))))
54	53	pm2.43i 53	. . . . . 6 ⊢ ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵 ⊆ 𝐴 → ((𝐴 ∖ 𝐵) ≠ ∅ → 𝐵 ∈ 𝐴)))
55	7, 54	syl7 75	. . . . 5 ⊢ ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵 ⊆ 𝐴 → ((𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴) → 𝐵 ∈ 𝐴)))
56	55	exp4a 437	. . . 4 ⊢ ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵 ⊆ 𝐴 → (𝐵 ⊆ 𝐴 → (𝐵 ≠ 𝐴 → 𝐵 ∈ 𝐴))))
57	56	pm2.43d 54	. . 3 ⊢ ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵 ⊆ 𝐴 → (𝐵 ≠ 𝐴 → 𝐵 ∈ 𝐴)))
58	57	impd 416	. 2 ⊢ ((Ord 𝐴 ∧ Tr 𝐵) → ((𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴) → 𝐵 ∈ 𝐴))
59	6, 58	impbid 215	1 ⊢ ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵 ∈ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ w3o 1102   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∃wrex 3087   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  Tr wtr 5212   E cep 5550   Fr wfr 5601   We wwe 5603  Ord word 6361

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-ord 6365

This theorem is used by:  ordelssne  6389  dfon2  36554