Theorem php3OLD 9226

Description: Obsolete version of php3 9214 as of 26-Nov-2024. (Contributed by NM, 22-Aug-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
php3OLD	⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴)

Proof of Theorem php3OLD

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

Step	Hyp	Ref	Expression
1		isfi 8974	. . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
2		relen 8946	. . . . . . . . 9 ⊢ Rel ≈
3	2	brrelex1i 5731	. . . . . . . 8 ⊢ (𝐴 ≈ 𝑥 → 𝐴 ∈ V)
4		pssss 4094	. . . . . . . 8 ⊢ (𝐵 ⊊ 𝐴 → 𝐵 ⊆ 𝐴)
5		ssdomg 8998	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝐵 ⊆ 𝐴 → 𝐵 ≼ 𝐴))
6	5	imp 405	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝐵 ⊆ 𝐴) → 𝐵 ≼ 𝐴)
7	3, 4, 6	syl2an 594	. . . . . . 7 ⊢ ((𝐴 ≈ 𝑥 ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≼ 𝐴)
8	7	adantll 710	. . . . . 6 ⊢ (((𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥) ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≼ 𝐴)
9		bren 8951	. . . . . . . . 9 ⊢ (𝐴 ≈ 𝑥 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝑥)
10		imass2 6100	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ⊆ 𝐴 → (𝑓 “ 𝐵) ⊆ (𝑓 “ 𝐴))
11	4, 10	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ⊊ 𝐴 → (𝑓 “ 𝐵) ⊆ (𝑓 “ 𝐴))
12	11	adantl 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → (𝑓 “ 𝐵) ⊆ (𝑓 “ 𝐴))
13		pssnel 4469	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ⊊ 𝐴 → ∃𝑦(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))
14		eldif 3957	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝐴 ∖ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))
15		f1ofn 6833	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → 𝑓 Fn 𝐴)
16		difss 4130	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
17		fnfvima 7236	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑓 Fn 𝐴 ∧ (𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ 𝑦 ∈ (𝐴 ∖ 𝐵)) → (𝑓‘𝑦) ∈ (𝑓 “ (𝐴 ∖ 𝐵)))
18	17	3expia 1119	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓 Fn 𝐴 ∧ (𝐴 ∖ 𝐵) ⊆ 𝐴) → (𝑦 ∈ (𝐴 ∖ 𝐵) → (𝑓‘𝑦) ∈ (𝑓 “ (𝐴 ∖ 𝐵))))
19	15, 16, 18	sylancl 584	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑦 ∈ (𝐴 ∖ 𝐵) → (𝑓‘𝑦) ∈ (𝑓 “ (𝐴 ∖ 𝐵))))
20		dff1o3 6838	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓:𝐴–1-1-onto→𝑥 ↔ (𝑓:𝐴–onto→𝑥 ∧ Fun ◡𝑓))
21		imadif 6631	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Fun ◡𝑓 → (𝑓 “ (𝐴 ∖ 𝐵)) = ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)))
22	20, 21	simplbiim 503	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑓 “ (𝐴 ∖ 𝐵)) = ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)))
23	22	eleq2d 2817	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → ((𝑓‘𝑦) ∈ (𝑓 “ (𝐴 ∖ 𝐵)) ↔ (𝑓‘𝑦) ∈ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵))))
24	19, 23	sylibd 238	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑦 ∈ (𝐴 ∖ 𝐵) → (𝑓‘𝑦) ∈ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵))))
25		n0i 4332	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓‘𝑦) ∈ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) → ¬ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅)
26	24, 25	syl6 35	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑦 ∈ (𝐴 ∖ 𝐵) → ¬ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅))
27	14, 26	biimtrrid 242	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵) → ¬ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅))
28	27	exlimdv 1934	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (∃𝑦(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵) → ¬ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅))
29	28	imp 405	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ ∃𝑦(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)) → ¬ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅)
30	13, 29	sylan2 591	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → ¬ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅)
31		ssdif0 4362	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 “ 𝐴) ⊆ (𝑓 “ 𝐵) ↔ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅)
32	30, 31	sylnibr 328	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → ¬ (𝑓 “ 𝐴) ⊆ (𝑓 “ 𝐵))
33		dfpss3 4085	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 “ 𝐵) ⊊ (𝑓 “ 𝐴) ↔ ((𝑓 “ 𝐵) ⊆ (𝑓 “ 𝐴) ∧ ¬ (𝑓 “ 𝐴) ⊆ (𝑓 “ 𝐵)))
34	12, 32, 33	sylanbrc 581	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → (𝑓 “ 𝐵) ⊊ (𝑓 “ 𝐴))
35		imadmrn 6068	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 “ dom 𝑓) = ran 𝑓
36		f1odm 6836	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → dom 𝑓 = 𝐴)
37	36	imaeq2d 6058	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑓 “ dom 𝑓) = (𝑓 “ 𝐴))
38		f1ofo 6839	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → 𝑓:𝐴–onto→𝑥)
39		forn 6807	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:𝐴–onto→𝑥 → ran 𝑓 = 𝑥)
40	38, 39	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → ran 𝑓 = 𝑥)
41	35, 37, 40	3eqtr3a 2794	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑓 “ 𝐴) = 𝑥)
42	41	psseq2d 4092	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → ((𝑓 “ 𝐵) ⊊ (𝑓 “ 𝐴) ↔ (𝑓 “ 𝐵) ⊊ 𝑥))
43	42	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → ((𝑓 “ 𝐵) ⊊ (𝑓 “ 𝐴) ↔ (𝑓 “ 𝐵) ⊊ 𝑥))
44	34, 43	mpbid 231	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → (𝑓 “ 𝐵) ⊊ 𝑥)
45		php 9212	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ω ∧ (𝑓 “ 𝐵) ⊊ 𝑥) → ¬ 𝑥 ≈ (𝑓 “ 𝐵))
46	44, 45	sylan2 591	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ω ∧ (𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴)) → ¬ 𝑥 ≈ (𝑓 “ 𝐵))
47		f1of1 6831	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → 𝑓:𝐴–1-1→𝑥)
48		f1ores 6846	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:𝐴–1-1→𝑥 ∧ 𝐵 ⊆ 𝐴) → (𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵))
49	47, 4, 48	syl2an 594	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → (𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵))
50		vex 3476	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑓 ∈ V
51	50	resex 6028	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ↾ 𝐵) ∈ V
52		f1oeq1 6820	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑓 ↾ 𝐵) → (𝑦:𝐵–1-1-onto→(𝑓 “ 𝐵) ↔ (𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵)))
53	51, 52	spcev 3595	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵) → ∃𝑦 𝑦:𝐵–1-1-onto→(𝑓 “ 𝐵))
54		bren 8951	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ≈ (𝑓 “ 𝐵) ↔ ∃𝑦 𝑦:𝐵–1-1-onto→(𝑓 “ 𝐵))
55	53, 54	sylibr 233	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵) → 𝐵 ≈ (𝑓 “ 𝐵))
56	49, 55	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≈ (𝑓 “ 𝐵))
57		entr 9004	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ≈ 𝐵 ∧ 𝐵 ≈ (𝑓 “ 𝐵)) → 𝑥 ≈ (𝑓 “ 𝐵))
58	57	expcom 412	. . . . . . . . . . . . . 14 ⊢ (𝐵 ≈ (𝑓 “ 𝐵) → (𝑥 ≈ 𝐵 → 𝑥 ≈ (𝑓 “ 𝐵)))
59	56, 58	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → (𝑥 ≈ 𝐵 → 𝑥 ≈ (𝑓 “ 𝐵)))
60	59	adantl 480	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ω ∧ (𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴)) → (𝑥 ≈ 𝐵 → 𝑥 ≈ (𝑓 “ 𝐵)))
61	46, 60	mtod 197	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ω ∧ (𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴)) → ¬ 𝑥 ≈ 𝐵)
62	61	exp32 419	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω → (𝑓:𝐴–1-1-onto→𝑥 → (𝐵 ⊊ 𝐴 → ¬ 𝑥 ≈ 𝐵)))
63	62	exlimdv 1934	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → (∃𝑓 𝑓:𝐴–1-1-onto→𝑥 → (𝐵 ⊊ 𝐴 → ¬ 𝑥 ≈ 𝐵)))
64	9, 63	biimtrid 241	. . . . . . . 8 ⊢ (𝑥 ∈ ω → (𝐴 ≈ 𝑥 → (𝐵 ⊊ 𝐴 → ¬ 𝑥 ≈ 𝐵)))
65	64	imp31 416	. . . . . . 7 ⊢ (((𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥) ∧ 𝐵 ⊊ 𝐴) → ¬ 𝑥 ≈ 𝐵)
66		entr 9004	. . . . . . . . . 10 ⊢ ((𝐵 ≈ 𝐴 ∧ 𝐴 ≈ 𝑥) → 𝐵 ≈ 𝑥)
67	66	ex 411	. . . . . . . . 9 ⊢ (𝐵 ≈ 𝐴 → (𝐴 ≈ 𝑥 → 𝐵 ≈ 𝑥))
68		ensym 9001	. . . . . . . . 9 ⊢ (𝐵 ≈ 𝑥 → 𝑥 ≈ 𝐵)
69	67, 68	syl6com 37	. . . . . . . 8 ⊢ (𝐴 ≈ 𝑥 → (𝐵 ≈ 𝐴 → 𝑥 ≈ 𝐵))
70	69	ad2antlr 723	. . . . . . 7 ⊢ (((𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥) ∧ 𝐵 ⊊ 𝐴) → (𝐵 ≈ 𝐴 → 𝑥 ≈ 𝐵))
71	65, 70	mtod 197	. . . . . 6 ⊢ (((𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥) ∧ 𝐵 ⊊ 𝐴) → ¬ 𝐵 ≈ 𝐴)
72		brsdom 8973	. . . . . 6 ⊢ (𝐵 ≺ 𝐴 ↔ (𝐵 ≼ 𝐴 ∧ ¬ 𝐵 ≈ 𝐴))
73	8, 71, 72	sylanbrc 581	. . . . 5 ⊢ (((𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥) ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴)
74	73	exp31 418	. . . 4 ⊢ (𝑥 ∈ ω → (𝐴 ≈ 𝑥 → (𝐵 ⊊ 𝐴 → 𝐵 ≺ 𝐴)))
75	74	rexlimiv 3146	. . 3 ⊢ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → (𝐵 ⊊ 𝐴 → 𝐵 ≺ 𝐴))
76	1, 75	sylbi 216	. 2 ⊢ (𝐴 ∈ Fin → (𝐵 ⊊ 𝐴 → 𝐵 ≺ 𝐴))
77	76	imp 405	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539  ∃wex 1779   ∈ wcel 2104  ∃wrex 3068  Vcvv 3472   ∖ cdif 3944   ⊆ wss 3947   ⊊ wpss 3948  ∅c0 4321   class class class wbr 5147  ^◡ccnv 5674  dom cdm 5675  ran crn 5676   ↾ cres 5677   “ cima 5678  Fun wfun 6536   Fn wfn 6537  –1-1→wf1 6539  –onto→wfo 6540  –1-1-onto→wf1o 6541  ‘cfv 6542  ωcom 7857   ≈ cen 8938   ≼ cdom 8939   ≺ csdm 8940  Fincfn 8941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-om 7858  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945

This theorem is referenced by: (None)