Theorem dfon2lem5 30936

Description: Lemma for dfon2 30941. 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 30935	. . 3 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
4		dfpss2 3654	. . . . . 6 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵))
5		dfpss2 3654	. . . . . . 7 ⊢ (𝐵 ⊊ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ ¬ 𝐵 = 𝐴))
6		eqcom 2617	. . . . . . . . 9 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
7	6	notbii 309	. . . . . . . 8 ⊢ (¬ 𝐵 = 𝐴 ↔ ¬ 𝐴 = 𝐵)
8	7	anbi2i 726	. . . . . . 7 ⊢ ((𝐵 ⊆ 𝐴 ∧ ¬ 𝐵 = 𝐴) ↔ (𝐵 ⊆ 𝐴 ∧ ¬ 𝐴 = 𝐵))
9	5, 8	bitri 263	. . . . . 6 ⊢ (𝐵 ⊊ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ ¬ 𝐴 = 𝐵))
10	4, 9	orbi12i 542	. . . . 5 ⊢ ((𝐴 ⊊ 𝐵 ∨ 𝐵 ⊊ 𝐴) ↔ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵) ∨ (𝐵 ⊆ 𝐴 ∧ ¬ 𝐴 = 𝐵)))
11		andir 908	. . . . 5 ⊢ (((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) ∧ ¬ 𝐴 = 𝐵) ↔ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵) ∨ (𝐵 ⊆ 𝐴 ∧ ¬ 𝐴 = 𝐵)))
12	10, 11	bitr4i 266	. . . 4 ⊢ ((𝐴 ⊊ 𝐵 ∨ 𝐵 ⊊ 𝐴) ↔ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) ∧ ¬ 𝐴 = 𝐵))
13		orcom 401	. . . . 5 ⊢ ((𝐴 ⊊ 𝐵 ∨ 𝐵 ⊊ 𝐴) ↔ (𝐵 ⊊ 𝐴 ∨ 𝐴 ⊊ 𝐵))
14		dfon2lem3 30934	. . . . . . . . 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 474	. . . . . . 7 ⊢ (∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵) → Tr 𝐵)
17		psseq1 3656	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (𝑥 ⊊ 𝐴 ↔ 𝐵 ⊊ 𝐴))
18		treq 4686	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (Tr 𝑥 ↔ Tr 𝐵))
19	17, 18	anbi12d 743	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → ((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) ↔ (𝐵 ⊊ 𝐴 ∧ Tr 𝐵)))
20		eleq1 2676	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
21	19, 20	imbi12d 333	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ↔ ((𝐵 ⊊ 𝐴 ∧ Tr 𝐵) → 𝐵 ∈ 𝐴)))
22	2, 21	spcv 3272	. . . . . . . . 9 ⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → ((𝐵 ⊊ 𝐴 ∧ Tr 𝐵) → 𝐵 ∈ 𝐴))
23	22	expcomd 453	. . . . . . . 8 ⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → (Tr 𝐵 → (𝐵 ⊊ 𝐴 → 𝐵 ∈ 𝐴)))
24	23	imp 444	. . . . . . 7 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ Tr 𝐵) → (𝐵 ⊊ 𝐴 → 𝐵 ∈ 𝐴))
25	16, 24	sylan2 490	. . . . . 6 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → (𝐵 ⊊ 𝐴 → 𝐵 ∈ 𝐴))
26		dfon2lem3 30934	. . . . . . . . 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 474	. . . . . . 7 ⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → Tr 𝐴)
29		psseq1 3656	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝑦 ⊊ 𝐵 ↔ 𝐴 ⊊ 𝐵))
30		treq 4686	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (Tr 𝑦 ↔ Tr 𝐴))
31	29, 30	anbi12d 743	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → ((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) ↔ (𝐴 ⊊ 𝐵 ∧ Tr 𝐴)))
32		eleq1 2676	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
33	31, 32	imbi12d 333	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵) ↔ ((𝐴 ⊊ 𝐵 ∧ Tr 𝐴) → 𝐴 ∈ 𝐵)))
34	1, 33	spcv 3272	. . . . . . . 8 ⊢ (∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵) → ((𝐴 ⊊ 𝐵 ∧ Tr 𝐴) → 𝐴 ∈ 𝐵))
35	34	expcomd 453	. . . . . . 7 ⊢ (∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵) → (Tr 𝐴 → (𝐴 ⊊ 𝐵 → 𝐴 ∈ 𝐵)))
36	28, 35	mpan9 485	. . . . . 6 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → (𝐴 ⊊ 𝐵 → 𝐴 ∈ 𝐵))
37	25, 36	orim12d 879	. . . . 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	syl5bi 231	. . . 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	syl5bir 232	. . 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 707	. 2 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → (¬ 𝐴 = 𝐵 → (𝐵 ∈ 𝐴 ∨ 𝐴 ∈ 𝐵)))
41		3orrot 1037	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ↔ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴 ∨ 𝐴 ∈ 𝐵))
42		3orass 1034	. . . 4 ⊢ ((𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴 ∨ 𝐴 ∈ 𝐵) ↔ (𝐴 = 𝐵 ∨ (𝐵 ∈ 𝐴 ∨ 𝐴 ∈ 𝐵)))
43		df-or 384	. . . 4 ⊢ ((𝐴 = 𝐵 ∨ (𝐵 ∈ 𝐴 ∨ 𝐴 ∈ 𝐵)) ↔ (¬ 𝐴 = 𝐵 → (𝐵 ∈ 𝐴 ∨ 𝐴 ∈ 𝐵)))
44	42, 43	bitri 263	. . 3 ⊢ ((𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴 ∨ 𝐴 ∈ 𝐵) ↔ (¬ 𝐴 = 𝐵 → (𝐵 ∈ 𝐴 ∨ 𝐴 ∈ 𝐵)))
45	41, 44	bitri 263	. 2 ⊢ ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ↔ (¬ 𝐴 = 𝐵 → (𝐵 ∈ 𝐴 ∨ 𝐴 ∈ 𝐵)))
46	40, 45	sylibr 223	1 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   ∨ w3o 1030  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540   ⊊ wpss 3541  Tr wtr 4680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-pw 4110  df-sn 4126  df-pr 4128  df-uni 4373  df-iun 4457  df-tr 4681  df-suc 5646

This theorem is referenced by:  dfon2lem6  30937  dfon2  30941