Theorem php3OLD 9238

Description: Obsolete version of php3 9228 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 8995	. . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
2		relen 8969	. . . . . . . . 9 ⊢ Rel ≈
3	2	brrelex1i 5715	. . . . . . . 8 ⊢ (𝐴 ≈ 𝑥 → 𝐴 ∈ V)
4		pssss 4078	. . . . . . . 8 ⊢ (𝐵 ⊊ 𝐴 → 𝐵 ⊆ 𝐴)
5		ssdomg 9019	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝐵 ⊆ 𝐴 → 𝐵 ≼ 𝐴))
6	5	imp 406	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝐵 ⊆ 𝐴) → 𝐵 ≼ 𝐴)
7	3, 4, 6	syl2an 596	. . . . . . 7 ⊢ ((𝐴 ≈ 𝑥 ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≼ 𝐴)
8	7	adantll 714	. . . . . 6 ⊢ (((𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥) ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≼ 𝐴)
9		bren 8974	. . . . . . . . 9 ⊢ (𝐴 ≈ 𝑥 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝑥)
10		imass2 6094	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ⊆ 𝐴 → (𝑓 “ 𝐵) ⊆ (𝑓 “ 𝐴))
11	4, 10	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ⊊ 𝐴 → (𝑓 “ 𝐵) ⊆ (𝑓 “ 𝐴))
12	11	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → (𝑓 “ 𝐵) ⊆ (𝑓 “ 𝐴))
13		pssnel 4451	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ⊊ 𝐴 → ∃𝑦(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))
14		eldif 3941	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝐴 ∖ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))
15		f1ofn 6824	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → 𝑓 Fn 𝐴)
16		difss 4116	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
17		fnfvima 7230	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑓 Fn 𝐴 ∧ (𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ 𝑦 ∈ (𝐴 ∖ 𝐵)) → (𝑓‘𝑦) ∈ (𝑓 “ (𝐴 ∖ 𝐵)))
18	17	3expia 1121	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓 Fn 𝐴 ∧ (𝐴 ∖ 𝐵) ⊆ 𝐴) → (𝑦 ∈ (𝐴 ∖ 𝐵) → (𝑓‘𝑦) ∈ (𝑓 “ (𝐴 ∖ 𝐵))))
19	15, 16, 18	sylancl 586	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑦 ∈ (𝐴 ∖ 𝐵) → (𝑓‘𝑦) ∈ (𝑓 “ (𝐴 ∖ 𝐵))))
20		dff1o3 6829	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓:𝐴–1-1-onto→𝑥 ↔ (𝑓:𝐴–onto→𝑥 ∧ Fun ◡𝑓))
21		imadif 6625	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Fun ◡𝑓 → (𝑓 “ (𝐴 ∖ 𝐵)) = ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)))
22	20, 21	simplbiim 504	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑓 “ (𝐴 ∖ 𝐵)) = ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)))
23	22	eleq2d 2821	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → ((𝑓‘𝑦) ∈ (𝑓 “ (𝐴 ∖ 𝐵)) ↔ (𝑓‘𝑦) ∈ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵))))
24	19, 23	sylibd 239	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑦 ∈ (𝐴 ∖ 𝐵) → (𝑓‘𝑦) ∈ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵))))
25		n0i 4320	. . . . . . . . . . . . . . . . . . . . 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 243	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵) → ¬ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅))
28	27	exlimdv 1933	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (∃𝑦(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵) → ¬ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅))
29	28	imp 406	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ ∃𝑦(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)) → ¬ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅)
30	13, 29	sylan2 593	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → ¬ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅)
31		ssdif0 4346	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 “ 𝐴) ⊆ (𝑓 “ 𝐵) ↔ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅)
32	30, 31	sylnibr 329	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → ¬ (𝑓 “ 𝐴) ⊆ (𝑓 “ 𝐵))
33		dfpss3 4069	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 “ 𝐵) ⊊ (𝑓 “ 𝐴) ↔ ((𝑓 “ 𝐵) ⊆ (𝑓 “ 𝐴) ∧ ¬ (𝑓 “ 𝐴) ⊆ (𝑓 “ 𝐵)))
34	12, 32, 33	sylanbrc 583	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → (𝑓 “ 𝐵) ⊊ (𝑓 “ 𝐴))
35		imadmrn 6062	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 “ dom 𝑓) = ran 𝑓
36		f1odm 6827	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → dom 𝑓 = 𝐴)
37	36	imaeq2d 6052	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑓 “ dom 𝑓) = (𝑓 “ 𝐴))
38		f1ofo 6830	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → 𝑓:𝐴–onto→𝑥)
39		forn 6798	. . . . . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑓 “ 𝐴) = 𝑥)
42	41	psseq2d 4076	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → ((𝑓 “ 𝐵) ⊊ (𝑓 “ 𝐴) ↔ (𝑓 “ 𝐵) ⊊ 𝑥))
43	42	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → ((𝑓 “ 𝐵) ⊊ (𝑓 “ 𝐴) ↔ (𝑓 “ 𝐵) ⊊ 𝑥))
44	34, 43	mpbid 232	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → (𝑓 “ 𝐵) ⊊ 𝑥)
45		php 9226	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ω ∧ (𝑓 “ 𝐵) ⊊ 𝑥) → ¬ 𝑥 ≈ (𝑓 “ 𝐵))
46	44, 45	sylan2 593	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ω ∧ (𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴)) → ¬ 𝑥 ≈ (𝑓 “ 𝐵))
47		f1of1 6822	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → 𝑓:𝐴–1-1→𝑥)
48		f1ores 6837	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:𝐴–1-1→𝑥 ∧ 𝐵 ⊆ 𝐴) → (𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵))
49	47, 4, 48	syl2an 596	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → (𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵))
50		vex 3468	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑓 ∈ V
51	50	resex 6021	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ↾ 𝐵) ∈ V
52		f1oeq1 6811	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑓 ↾ 𝐵) → (𝑦:𝐵–1-1-onto→(𝑓 “ 𝐵) ↔ (𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵)))
53	51, 52	spcev 3590	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵) → ∃𝑦 𝑦:𝐵–1-1-onto→(𝑓 “ 𝐵))
54		bren 8974	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ≈ (𝑓 “ 𝐵) ↔ ∃𝑦 𝑦:𝐵–1-1-onto→(𝑓 “ 𝐵))
55	53, 54	sylibr 234	. . . . . . . . . . . . . . 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 9025	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ≈ 𝐵 ∧ 𝐵 ≈ (𝑓 “ 𝐵)) → 𝑥 ≈ (𝑓 “ 𝐵))
58	57	expcom 413	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ω ∧ (𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴)) → (𝑥 ≈ 𝐵 → 𝑥 ≈ (𝑓 “ 𝐵)))
61	46, 60	mtod 198	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ω ∧ (𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴)) → ¬ 𝑥 ≈ 𝐵)
62	61	exp32 420	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω → (𝑓:𝐴–1-1-onto→𝑥 → (𝐵 ⊊ 𝐴 → ¬ 𝑥 ≈ 𝐵)))
63	62	exlimdv 1933	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → (∃𝑓 𝑓:𝐴–1-1-onto→𝑥 → (𝐵 ⊊ 𝐴 → ¬ 𝑥 ≈ 𝐵)))
64	9, 63	biimtrid 242	. . . . . . . 8 ⊢ (𝑥 ∈ ω → (𝐴 ≈ 𝑥 → (𝐵 ⊊ 𝐴 → ¬ 𝑥 ≈ 𝐵)))
65	64	imp31 417	. . . . . . 7 ⊢ (((𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥) ∧ 𝐵 ⊊ 𝐴) → ¬ 𝑥 ≈ 𝐵)
66		entr 9025	. . . . . . . . . 10 ⊢ ((𝐵 ≈ 𝐴 ∧ 𝐴 ≈ 𝑥) → 𝐵 ≈ 𝑥)
67	66	ex 412	. . . . . . . . 9 ⊢ (𝐵 ≈ 𝐴 → (𝐴 ≈ 𝑥 → 𝐵 ≈ 𝑥))
68		ensym 9022	. . . . . . . . 9 ⊢ (𝐵 ≈ 𝑥 → 𝑥 ≈ 𝐵)
69	67, 68	syl6com 37	. . . . . . . 8 ⊢ (𝐴 ≈ 𝑥 → (𝐵 ≈ 𝐴 → 𝑥 ≈ 𝐵))
70	69	ad2antlr 727	. . . . . . 7 ⊢ (((𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥) ∧ 𝐵 ⊊ 𝐴) → (𝐵 ≈ 𝐴 → 𝑥 ≈ 𝐵))
71	65, 70	mtod 198	. . . . . 6 ⊢ (((𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥) ∧ 𝐵 ⊊ 𝐴) → ¬ 𝐵 ≈ 𝐴)
72		brsdom 8994	. . . . . 6 ⊢ (𝐵 ≺ 𝐴 ↔ (𝐵 ≼ 𝐴 ∧ ¬ 𝐵 ≈ 𝐴))
73	8, 71, 72	sylanbrc 583	. . . . 5 ⊢ (((𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥) ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴)
74	73	exp31 419	. . . 4 ⊢ (𝑥 ∈ ω → (𝐴 ≈ 𝑥 → (𝐵 ⊊ 𝐴 → 𝐵 ≺ 𝐴)))
75	74	rexlimiv 3135	. . 3 ⊢ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → (𝐵 ⊊ 𝐴 → 𝐵 ≺ 𝐴))
76	1, 75	sylbi 217	. 2 ⊢ (𝐴 ∈ Fin → (𝐵 ⊊ 𝐴 → 𝐵 ≺ 𝐴))
77	76	imp 406	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃wrex 3061  Vcvv 3464   ∖ cdif 3928   ⊆ wss 3931   ⊊ wpss 3932  ∅c0 4313   class class class wbr 5124  ^◡ccnv 5658  dom cdm 5659  ran crn 5660   ↾ cres 5661   “ cima 5662  Fun wfun 6530   Fn wfn 6531  –1-1→wf1 6533  –onto→wfo 6534  –1-1-onto→wf1o 6535  ‘cfv 6536  ωcom 7866   ≈ cen 8961   ≼ cdom 8962   ≺ csdm 8963  Fincfn 8964

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-om 7867  df-1o 8485  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968

This theorem is referenced by: (None)