Theorem dfon2lem5 36529

Description: Lemma for dfon2 36534. Two sets satisfying the new definition also satisfy trichotomy with respect to ∈. (Contributed by Scott Fenton, 25-Feb-2011.)

Hypotheses

Ref	Expression
dfon2lem5.1	⊢ 𝐴 ∈ V
dfon2lem5.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
dfon2lem5	⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Proof of Theorem dfon2lem5

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfon2lem5.1	. . . 4 ⊢ 𝐴 ∈ V
2		dfon2lem5.2	. . . 4 ⊢ 𝐵 ∈ V
3	1, 2	dfon2lem4 36528	. . 3 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
4		dfpss2 4036	. . . . . 6 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵))
5		dfpss2 4036	. . . . . . 7 ⊢ (𝐵 ⊊ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ ¬ 𝐵 = 𝐴))
6		eqcom 2768	. . . . . . . . 9 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
7	6	notbii 323	. . . . . . . 8 ⊢ (¬ 𝐵 = 𝐴 ↔ ¬ 𝐴 = 𝐵)
8	7	anbi2i 635	. . . . . . 7 ⊢ ((𝐵 ⊆ 𝐴 ∧ ¬ 𝐵 = 𝐴) ↔ (𝐵 ⊆ 𝐴 ∧ ¬ 𝐴 = 𝐵))
9	5, 8	bitri 278	. . . . . 6 ⊢ (𝐵 ⊊ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ ¬ 𝐴 = 𝐵))
10	4, 9	orbi12i 928	. . . . 5 ⊢ ((𝐴 ⊊ 𝐵 ∨ 𝐵 ⊊ 𝐴) ↔ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵) ∨ (𝐵 ⊆ 𝐴 ∧ ¬ 𝐴 = 𝐵)))
11		andir 1026	. . . . 5 ⊢ (((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) ∧ ¬ 𝐴 = 𝐵) ↔ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵) ∨ (𝐵 ⊆ 𝐴 ∧ ¬ 𝐴 = 𝐵)))
12	10, 11	bitr4i 281	. . . 4 ⊢ ((𝐴 ⊊ 𝐵 ∨ 𝐵 ⊊ 𝐴) ↔ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) ∧ ¬ 𝐴 = 𝐵))
13		orcom 884	. . . . 5 ⊢ ((𝐴 ⊊ 𝐵 ∨ 𝐵 ⊊ 𝐴) ↔ (𝐵 ⊊ 𝐴 ∨ 𝐴 ⊊ 𝐵))
14		dfon2lem3 36527	. . . . . . . . 9 ⊢ (𝐵 ∈ V → (∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵) → (Tr 𝐵 ∧ ∀𝑧 ∈ 𝐵 ¬ 𝑧 ∈ 𝑧)))
15	2, 14	ax-mp 5	. . . . . . . 8 ⊢ (∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵) → (Tr 𝐵 ∧ ∀𝑧 ∈ 𝐵 ¬ 𝑧 ∈ 𝑧))
16	15	simpld 500	. . . . . . 7 ⊢ (∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵) → Tr 𝐵)
17		psseq1 4038	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (𝑥 ⊊ 𝐴 ↔ 𝐵 ⊊ 𝐴))
18		treq 5219	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (Tr 𝑥 ↔ Tr 𝐵))
19	17, 18	anbi12d 644	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → ((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) ↔ (𝐵 ⊊ 𝐴 ∧ Tr 𝐵)))
20		eleq1 2849	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
21	19, 20	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ↔ ((𝐵 ⊊ 𝐴 ∧ Tr 𝐵) → 𝐵 ∈ 𝐴)))
22	2, 21	spcv 3560	. . . . . . . . 9 ⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → ((𝐵 ⊊ 𝐴 ∧ Tr 𝐵) → 𝐵 ∈ 𝐴))
23	22	expcomd 422	. . . . . . . 8 ⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → (Tr 𝐵 → (𝐵 ⊊ 𝐴 → 𝐵 ∈ 𝐴)))
24	23	imp 412	. . . . . . 7 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ Tr 𝐵) → (𝐵 ⊊ 𝐴 → 𝐵 ∈ 𝐴))
25	16, 24	sylan2 605	. . . . . 6 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → (𝐵 ⊊ 𝐴 → 𝐵 ∈ 𝐴))
26		dfon2lem3 36527	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → (Tr 𝐴 ∧ ∀𝑧 ∈ 𝐴 ¬ 𝑧 ∈ 𝑧)))
27	1, 26	ax-mp 5	. . . . . . . 8 ⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → (Tr 𝐴 ∧ ∀𝑧 ∈ 𝐴 ¬ 𝑧 ∈ 𝑧))
28	27	simpld 500	. . . . . . 7 ⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → Tr 𝐴)
29		psseq1 4038	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝑦 ⊊ 𝐵 ↔ 𝐴 ⊊ 𝐵))
30		treq 5219	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (Tr 𝑦 ↔ Tr 𝐴))
31	29, 30	anbi12d 644	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → ((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) ↔ (𝐴 ⊊ 𝐵 ∧ Tr 𝐴)))
32		eleq1 2849	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
33	31, 32	imbi12d 347	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵) ↔ ((𝐴 ⊊ 𝐵 ∧ Tr 𝐴) → 𝐴 ∈ 𝐵)))
34	1, 33	spcv 3560	. . . . . . . 8 ⊢ (∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵) → ((𝐴 ⊊ 𝐵 ∧ Tr 𝐴) → 𝐴 ∈ 𝐵))
35	34	expcomd 422	. . . . . . 7 ⊢ (∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵) → (Tr 𝐴 → (𝐴 ⊊ 𝐵 → 𝐴 ∈ 𝐵)))
36	28, 35	mpan9 516	. . . . . 6 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → (𝐴 ⊊ 𝐵 → 𝐴 ∈ 𝐵))
37	25, 36	orim12d 979	. . . . 5 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → ((𝐵 ⊊ 𝐴 ∨ 𝐴 ⊊ 𝐵) → (𝐵 ∈ 𝐴 ∨ 𝐴 ∈ 𝐵)))
38	13, 37	biimtrid 245	. . . 4 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → ((𝐴 ⊊ 𝐵 ∨ 𝐵 ⊊ 𝐴) → (𝐵 ∈ 𝐴 ∨ 𝐴 ∈ 𝐵)))
39	12, 38	biimtrrid 246	. . 3 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → (((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) ∧ ¬ 𝐴 = 𝐵) → (𝐵 ∈ 𝐴 ∨ 𝐴 ∈ 𝐵)))
40	3, 39	mpand 708	. 2 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → (¬ 𝐴 = 𝐵 → (𝐵 ∈ 𝐴 ∨ 𝐴 ∈ 𝐵)))
41		3orrot 1108	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ↔ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴 ∨ 𝐴 ∈ 𝐵))
42		3orass 1106	. . . 4 ⊢ ((𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴 ∨ 𝐴 ∈ 𝐵) ↔ (𝐴 = 𝐵 ∨ (𝐵 ∈ 𝐴 ∨ 𝐴 ∈ 𝐵)))
43		df-or 862	. . . 4 ⊢ ((𝐴 = 𝐵 ∨ (𝐵 ∈ 𝐴 ∨ 𝐴 ∈ 𝐵)) ↔ (¬ 𝐴 = 𝐵 → (𝐵 ∈ 𝐴 ∨ 𝐴 ∈ 𝐵)))
44	42, 43	bitri 278	. . 3 ⊢ ((𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴 ∨ 𝐴 ∈ 𝐵) ↔ (¬ 𝐴 = 𝐵 → (𝐵 ∈ 𝐴 ∨ 𝐴 ∈ 𝐵)))
45	41, 44	bitri 278	. 2 ⊢ ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ↔ (¬ 𝐴 = 𝐵 → (𝐵 ∈ 𝐴 ∨ 𝐴 ∈ 𝐵)))
46	40, 45	sylibr 237	1 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∨ w3o 1102  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Vcvv 3451   ⊆ wss 3899   ⊊ wpss 3900  Tr wtr 5212

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-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-pw 4559  df-sn 4585  df-pr 4587  df-uni 4868  df-iun 4953  df-tr 5213  df-suc 6367

This theorem is used by:  dfon2lem6  36530  dfon2  36534