Theorem tz7.7 5749

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 5737	. . . 4 ⊢ (Ord 𝐴 → Tr 𝐴)
2		ordfr 5738	. . . 4 ⊢ (Ord 𝐴 → E Fr 𝐴)
3		tz7.2 5098	. . . . 5 ⊢ ((Tr 𝐴 ∧ E Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴))
4	3	3exp 1264	. . . 4 ⊢ (Tr 𝐴 → ( E Fr 𝐴 → (𝐵 ∈ 𝐴 → (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴))))
5	1, 2, 4	sylc 65	. . 3 ⊢ (Ord 𝐴 → (𝐵 ∈ 𝐴 → (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴)))
6	5	adantr 481	. 2 ⊢ ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵 ∈ 𝐴 → (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴)))
7		pssdifn0 3944	. . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴) → (𝐴 ∖ 𝐵) ≠ ∅)
8		difss 3737	. . . . . . . . . . . 12 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
9		tz7.5 5744	. . . . . . . . . . . 12 ⊢ ((Ord 𝐴 ∧ (𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)((𝐴 ∖ 𝐵) ∩ 𝑥) = ∅)
10	8, 9	mp3an2 1412	. . . . . . . . . . 11 ⊢ ((Ord 𝐴 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)((𝐴 ∖ 𝐵) ∩ 𝑥) = ∅)
11		eldifi 3732	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐴)
12		trss 4761	. . . . . . . . . . . . . . . . . 18 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
13		difin0ss 3946	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∖ 𝐵) ∩ 𝑥) = ∅ → (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵))
14	13	com12 32	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ⊆ 𝐴 → (((𝐴 ∖ 𝐵) ∩ 𝑥) = ∅ → 𝑥 ⊆ 𝐵))
15	11, 12, 14	syl56 36	. . . . . . . . . . . . . . . . 17 ⊢ (Tr 𝐴 → (𝑥 ∈ (𝐴 ∖ 𝐵) → (((𝐴 ∖ 𝐵) ∩ 𝑥) = ∅ → 𝑥 ⊆ 𝐵)))
16	1, 15	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (Ord 𝐴 → (𝑥 ∈ (𝐴 ∖ 𝐵) → (((𝐴 ∖ 𝐵) ∩ 𝑥) = ∅ → 𝑥 ⊆ 𝐵)))
17	16	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ (((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ (𝐴 ∖ 𝐵) → (((𝐴 ∖ 𝐵) ∩ 𝑥) = ∅ → 𝑥 ⊆ 𝐵)))
18	17	imp32 449	. . . . . . . . . . . . . 14 ⊢ ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ (𝐴 ∖ 𝐵) ∧ ((𝐴 ∖ 𝐵) ∩ 𝑥) = ∅)) → 𝑥 ⊆ 𝐵)
19		eleq1 2689	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
20	19	biimpcd 239	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ 𝐵 → (𝑦 = 𝑥 → 𝑥 ∈ 𝐵))
21		eldifn 3733	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ¬ 𝑥 ∈ 𝐵)
22	20, 21	nsyli 155	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ 𝐵 → (𝑥 ∈ (𝐴 ∖ 𝐵) → ¬ 𝑦 = 𝑥))
23	22	imp 445	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) → ¬ 𝑦 = 𝑥)
24	23	adantll 750	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) → ¬ 𝑦 = 𝑥)
25	24	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((Ord 𝐴 ∧ Tr 𝐵) ∧ ((𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵))) → ¬ 𝑦 = 𝑥)
26		trel 4759	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (Tr 𝐵 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵))
27	26	expcomd 454	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (Tr 𝐵 → (𝑦 ∈ 𝐵 → (𝑥 ∈ 𝑦 → 𝑥 ∈ 𝐵)))
28	27	imp 445	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((Tr 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ 𝑦 → 𝑥 ∈ 𝐵))
29	28, 21	nsyli 155	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((Tr 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ (𝐴 ∖ 𝐵) → ¬ 𝑥 ∈ 𝑦))
30	29	ex 450	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Tr 𝐵 → (𝑦 ∈ 𝐵 → (𝑥 ∈ (𝐴 ∖ 𝐵) → ¬ 𝑥 ∈ 𝑦)))
31	30	adantld 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Tr 𝐵 → ((𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ (𝐴 ∖ 𝐵) → ¬ 𝑥 ∈ 𝑦)))
32	31	imp32 449	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Tr 𝐵 ∧ ((𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵))) → ¬ 𝑥 ∈ 𝑦)
33	32	adantll 750	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((Ord 𝐴 ∧ Tr 𝐵) ∧ ((𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵))) → ¬ 𝑥 ∈ 𝑦)
34		ordwe 5736	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Ord 𝐴 → E We 𝐴)
35		ssel2 3598	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐴)
36	35, 11	anim12i 590	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) → (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
37		wecmpep 5106	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (( E We 𝐴 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → (𝑦 ∈ 𝑥 ∨ 𝑦 = 𝑥 ∨ 𝑥 ∈ 𝑦))
38	34, 36, 37	syl2an 494	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Ord 𝐴 ∧ ((𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵))) → (𝑦 ∈ 𝑥 ∨ 𝑦 = 𝑥 ∨ 𝑥 ∈ 𝑦))
39	38	adantlr 751	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((Ord 𝐴 ∧ Tr 𝐵) ∧ ((𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵))) → (𝑦 ∈ 𝑥 ∨ 𝑦 = 𝑥 ∨ 𝑥 ∈ 𝑦))
40	25, 33, 39	ecase23d 1436	. . . . . . . . . . . . . . . . . . 19 ⊢ (((Ord 𝐴 ∧ Tr 𝐵) ∧ ((𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵))) → 𝑦 ∈ 𝑥)
41	40	exp44 641	. . . . . . . . . . . . . . . . . 18 ⊢ ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵 ⊆ 𝐴 → (𝑦 ∈ 𝐵 → (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑦 ∈ 𝑥))))
42	41	com34 91	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵 ⊆ 𝐴 → (𝑥 ∈ (𝐴 ∖ 𝐵) → (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝑥))))
43	42	imp31 448	. . . . . . . . . . . . . . . 16 ⊢ ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) → (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝑥))
44	43	ssrdv 3609	. . . . . . . . . . . . . . 15 ⊢ ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) → 𝐵 ⊆ 𝑥)
45	44	adantrr 753	. . . . . . . . . . . . . 14 ⊢ ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ (𝐴 ∖ 𝐵) ∧ ((𝐴 ∖ 𝐵) ∩ 𝑥) = ∅)) → 𝐵 ⊆ 𝑥)
46	18, 45	eqssd 3620	. . . . . . . . . . . . 13 ⊢ ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ (𝐴 ∖ 𝐵) ∧ ((𝐴 ∖ 𝐵) ∩ 𝑥) = ∅)) → 𝑥 = 𝐵)
47	11	ad2antrl 764	. . . . . . . . . . . . 13 ⊢ ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ (𝐴 ∖ 𝐵) ∧ ((𝐴 ∖ 𝐵) ∩ 𝑥) = ∅)) → 𝑥 ∈ 𝐴)
48	46, 47	eqeltrrd 2702	. . . . . . . . . . . 12 ⊢ ((((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ (𝐴 ∖ 𝐵) ∧ ((𝐴 ∖ 𝐵) ∩ 𝑥) = ∅)) → 𝐵 ∈ 𝐴)
49	48	rexlimdvaa 3032	. . . . . . . . . . 11 ⊢ (((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵 ⊆ 𝐴) → (∃𝑥 ∈ (𝐴 ∖ 𝐵)((𝐴 ∖ 𝐵) ∩ 𝑥) = ∅ → 𝐵 ∈ 𝐴))
50	10, 49	syl5 34	. . . . . . . . . 10 ⊢ (((Ord 𝐴 ∧ Tr 𝐵) ∧ 𝐵 ⊆ 𝐴) → ((Ord 𝐴 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → 𝐵 ∈ 𝐴))
51	50	exp4b 632	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵 ⊆ 𝐴 → (Ord 𝐴 → ((𝐴 ∖ 𝐵) ≠ ∅ → 𝐵 ∈ 𝐴))))
52	51	com23 86	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ Tr 𝐵) → (Ord 𝐴 → (𝐵 ⊆ 𝐴 → ((𝐴 ∖ 𝐵) ≠ ∅ → 𝐵 ∈ 𝐴))))
53	52	adantrd 484	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ Tr 𝐵) → ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵 ⊆ 𝐴 → ((𝐴 ∖ 𝐵) ≠ ∅ → 𝐵 ∈ 𝐴))))
54	53	pm2.43i 52	. . . . . 6 ⊢ ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵 ⊆ 𝐴 → ((𝐴 ∖ 𝐵) ≠ ∅ → 𝐵 ∈ 𝐴)))
55	7, 54	syl7 74	. . . . 5 ⊢ ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵 ⊆ 𝐴 → ((𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴) → 𝐵 ∈ 𝐴)))
56	55	exp4a 633	. . . 4 ⊢ ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵 ⊆ 𝐴 → (𝐵 ⊆ 𝐴 → (𝐵 ≠ 𝐴 → 𝐵 ∈ 𝐴))))
57	56	pm2.43d 53	. . 3 ⊢ ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵 ⊆ 𝐴 → (𝐵 ≠ 𝐴 → 𝐵 ∈ 𝐴)))
58	57	impd 447	. 2 ⊢ ((Ord 𝐴 ∧ Tr 𝐵) → ((𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴) → 𝐵 ∈ 𝐴))
59	6, 58	impbid 202	1 ⊢ ((Ord 𝐴 ∧ Tr 𝐵) → (𝐵 ∈ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∨ w3o 1036   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913   ∖ cdif 3571   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  Tr wtr 4752   E cep 5028   Fr wfr 5070   We wwe 5072  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-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:  ordelssne  5750  dfon2  31697