Theorem php3 9135

Description: Corollary of Pigeonhole Principle. If 𝐴 is finite and 𝐵 is a proper subset of 𝐴, the 𝐵 is strictly less numerous than 𝐴. Stronger version of Corollary 6C of [Enderton] p. 135. (Contributed by NM, 22-Aug-2008.) Avoid ax-pow 5310. (Revised by BTernaryTau, 26-Nov-2024.)

Assertion

Ref	Expression
php3	⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴)

Proof of Theorem php3

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

Step	Hyp	Ref	Expression
1		isfi 8914	. . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
2		bren 8895	. . . . . 6 ⊢ (𝐴 ≈ 𝑥 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝑥)
3		pssss 4050	. . . . . . . . . . . . . 14 ⊢ (𝐵 ⊊ 𝐴 → 𝐵 ⊆ 𝐴)
4		imass2 6061	. . . . . . . . . . . . . 14 ⊢ (𝐵 ⊆ 𝐴 → (𝑓 “ 𝐵) ⊆ (𝑓 “ 𝐴))
5	3, 4	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐵 ⊊ 𝐴 → (𝑓 “ 𝐵) ⊆ (𝑓 “ 𝐴))
6	5	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → (𝑓 “ 𝐵) ⊆ (𝑓 “ 𝐴))
7		pssnel 4423	. . . . . . . . . . . . . 14 ⊢ (𝐵 ⊊ 𝐴 → ∃𝑦(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))
8		eldif 3911	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (𝐴 ∖ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))
9		f1ofn 6775	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → 𝑓 Fn 𝐴)
10		difss 4088	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
11		fnfvima 7179	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 Fn 𝐴 ∧ (𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ 𝑦 ∈ (𝐴 ∖ 𝐵)) → (𝑓‘𝑦) ∈ (𝑓 “ (𝐴 ∖ 𝐵)))
12	11	3expia 1121	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 Fn 𝐴 ∧ (𝐴 ∖ 𝐵) ⊆ 𝐴) → (𝑦 ∈ (𝐴 ∖ 𝐵) → (𝑓‘𝑦) ∈ (𝑓 “ (𝐴 ∖ 𝐵))))
13	9, 10, 12	sylancl 586	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑦 ∈ (𝐴 ∖ 𝐵) → (𝑓‘𝑦) ∈ (𝑓 “ (𝐴 ∖ 𝐵))))
14		dff1o3 6780	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓:𝐴–1-1-onto→𝑥 ↔ (𝑓:𝐴–onto→𝑥 ∧ Fun ◡𝑓))
15		imadif 6576	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Fun ◡𝑓 → (𝑓 “ (𝐴 ∖ 𝐵)) = ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)))
16	14, 15	simplbiim 504	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑓 “ (𝐴 ∖ 𝐵)) = ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)))
17	16	eleq2d 2822	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → ((𝑓‘𝑦) ∈ (𝑓 “ (𝐴 ∖ 𝐵)) ↔ (𝑓‘𝑦) ∈ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵))))
18	13, 17	sylibd 239	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑦 ∈ (𝐴 ∖ 𝐵) → (𝑓‘𝑦) ∈ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵))))
19		n0i 4292	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓‘𝑦) ∈ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) → ¬ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅)
20	18, 19	syl6 35	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑦 ∈ (𝐴 ∖ 𝐵) → ¬ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅))
21	8, 20	biimtrrid 243	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵) → ¬ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅))
22	21	exlimdv 1934	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (∃𝑦(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵) → ¬ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅))
23	22	imp 406	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ ∃𝑦(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)) → ¬ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅)
24	7, 23	sylan2 593	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → ¬ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅)
25		ssdif0 4318	. . . . . . . . . . . . 13 ⊢ ((𝑓 “ 𝐴) ⊆ (𝑓 “ 𝐵) ↔ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅)
26	24, 25	sylnibr 329	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → ¬ (𝑓 “ 𝐴) ⊆ (𝑓 “ 𝐵))
27		dfpss3 4041	. . . . . . . . . . . 12 ⊢ ((𝑓 “ 𝐵) ⊊ (𝑓 “ 𝐴) ↔ ((𝑓 “ 𝐵) ⊆ (𝑓 “ 𝐴) ∧ ¬ (𝑓 “ 𝐴) ⊆ (𝑓 “ 𝐵)))
28	6, 26, 27	sylanbrc 583	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → (𝑓 “ 𝐵) ⊊ (𝑓 “ 𝐴))
29		imadmrn 6029	. . . . . . . . . . . . . 14 ⊢ (𝑓 “ dom 𝑓) = ran 𝑓
30		f1odm 6778	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → dom 𝑓 = 𝐴)
31	30	imaeq2d 6019	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑓 “ dom 𝑓) = (𝑓 “ 𝐴))
32		f1ofo 6781	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → 𝑓:𝐴–onto→𝑥)
33		forn 6749	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐴–onto→𝑥 → ran 𝑓 = 𝑥)
34	32, 33	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → ran 𝑓 = 𝑥)
35	29, 31, 34	3eqtr3a 2795	. . . . . . . . . . . . 13 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑓 “ 𝐴) = 𝑥)
36	35	psseq2d 4048	. . . . . . . . . . . 12 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → ((𝑓 “ 𝐵) ⊊ (𝑓 “ 𝐴) ↔ (𝑓 “ 𝐵) ⊊ 𝑥))
37	36	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → ((𝑓 “ 𝐵) ⊊ (𝑓 “ 𝐴) ↔ (𝑓 “ 𝐵) ⊊ 𝑥))
38	28, 37	mpbid 232	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → (𝑓 “ 𝐵) ⊊ 𝑥)
39		php2 9134	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ω ∧ (𝑓 “ 𝐵) ⊊ 𝑥) → (𝑓 “ 𝐵) ≺ 𝑥)
40	38, 39	sylan2 593	. . . . . . . . 9 ⊢ ((𝑥 ∈ ω ∧ (𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴)) → (𝑓 “ 𝐵) ≺ 𝑥)
41		nnfi 9094	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω → 𝑥 ∈ Fin)
42		f1of1 6773	. . . . . . . . . . . 12 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → 𝑓:𝐴–1-1→𝑥)
43		f1ores 6788	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐴–1-1→𝑥 ∧ 𝐵 ⊆ 𝐴) → (𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵))
44	42, 3, 43	syl2an 596	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → (𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵))
45		vex 3444	. . . . . . . . . . . . . 14 ⊢ 𝑓 ∈ V
46	45	resex 5988	. . . . . . . . . . . . 13 ⊢ (𝑓 ↾ 𝐵) ∈ V
47		f1oeq1 6762	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑓 ↾ 𝐵) → (𝑦:𝐵–1-1-onto→(𝑓 “ 𝐵) ↔ (𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵)))
48	46, 47	spcev 3560	. . . . . . . . . . . 12 ⊢ ((𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵) → ∃𝑦 𝑦:𝐵–1-1-onto→(𝑓 “ 𝐵))
49		bren 8895	. . . . . . . . . . . 12 ⊢ (𝐵 ≈ (𝑓 “ 𝐵) ↔ ∃𝑦 𝑦:𝐵–1-1-onto→(𝑓 “ 𝐵))
50	48, 49	sylibr 234	. . . . . . . . . . 11 ⊢ ((𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵) → 𝐵 ≈ (𝑓 “ 𝐵))
51	44, 50	syl 17	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≈ (𝑓 “ 𝐵))
52		endom 8918	. . . . . . . . . . . 12 ⊢ (𝐵 ≈ (𝑓 “ 𝐵) → 𝐵 ≼ (𝑓 “ 𝐵))
53		sdomdom 8919	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 “ 𝐵) ≺ 𝑥 → (𝑓 “ 𝐵) ≼ 𝑥)
54		domfi 9115	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Fin ∧ (𝑓 “ 𝐵) ≼ 𝑥) → (𝑓 “ 𝐵) ∈ Fin)
55	53, 54	sylan2 593	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Fin ∧ (𝑓 “ 𝐵) ≺ 𝑥) → (𝑓 “ 𝐵) ∈ Fin)
56	55	3adant2 1131	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≼ (𝑓 “ 𝐵) ∧ (𝑓 “ 𝐵) ≺ 𝑥) → (𝑓 “ 𝐵) ∈ Fin)
57		domfi 9115	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 “ 𝐵) ∈ Fin ∧ 𝐵 ≼ (𝑓 “ 𝐵)) → 𝐵 ∈ Fin)
58	57	3adant3 1132	. . . . . . . . . . . . . 14 ⊢ (((𝑓 “ 𝐵) ∈ Fin ∧ 𝐵 ≼ (𝑓 “ 𝐵) ∧ (𝑓 “ 𝐵) ≺ 𝑥) → 𝐵 ∈ Fin)
59		domsdomtrfi 9128	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ Fin ∧ 𝐵 ≼ (𝑓 “ 𝐵) ∧ (𝑓 “ 𝐵) ≺ 𝑥) → 𝐵 ≺ 𝑥)
60	58, 59	syld3an1 1412	. . . . . . . . . . . . 13 ⊢ (((𝑓 “ 𝐵) ∈ Fin ∧ 𝐵 ≼ (𝑓 “ 𝐵) ∧ (𝑓 “ 𝐵) ≺ 𝑥) → 𝐵 ≺ 𝑥)
61	56, 60	syld3an1 1412	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≼ (𝑓 “ 𝐵) ∧ (𝑓 “ 𝐵) ≺ 𝑥) → 𝐵 ≺ 𝑥)
62	52, 61	syl3an2 1164	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≈ (𝑓 “ 𝐵) ∧ (𝑓 “ 𝐵) ≺ 𝑥) → 𝐵 ≺ 𝑥)
63	62	3expia 1121	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≈ (𝑓 “ 𝐵)) → ((𝑓 “ 𝐵) ≺ 𝑥 → 𝐵 ≺ 𝑥))
64	41, 51, 63	syl2an 596	. . . . . . . . 9 ⊢ ((𝑥 ∈ ω ∧ (𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴)) → ((𝑓 “ 𝐵) ≺ 𝑥 → 𝐵 ≺ 𝑥))
65	40, 64	mpd 15	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ (𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴)) → 𝐵 ≺ 𝑥)
66	65	exp32 420	. . . . . . 7 ⊢ (𝑥 ∈ ω → (𝑓:𝐴–1-1-onto→𝑥 → (𝐵 ⊊ 𝐴 → 𝐵 ≺ 𝑥)))
67	66	exlimdv 1934	. . . . . 6 ⊢ (𝑥 ∈ ω → (∃𝑓 𝑓:𝐴–1-1-onto→𝑥 → (𝐵 ⊊ 𝐴 → 𝐵 ≺ 𝑥)))
68	2, 67	biimtrid 242	. . . . 5 ⊢ (𝑥 ∈ ω → (𝐴 ≈ 𝑥 → (𝐵 ⊊ 𝐴 → 𝐵 ≺ 𝑥)))
69		ensymfib 9110	. . . . . . . . . . 11 ⊢ (𝑥 ∈ Fin → (𝑥 ≈ 𝐴 ↔ 𝐴 ≈ 𝑥))
70	69	adantr 480	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≺ 𝑥) → (𝑥 ≈ 𝐴 ↔ 𝐴 ≈ 𝑥))
71	70	biimp3ar 1472	. . . . . . . . 9 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≺ 𝑥 ∧ 𝐴 ≈ 𝑥) → 𝑥 ≈ 𝐴)
72		endom 8918	. . . . . . . . . 10 ⊢ (𝑥 ≈ 𝐴 → 𝑥 ≼ 𝐴)
73		sdomdom 8919	. . . . . . . . . . . . 13 ⊢ (𝐵 ≺ 𝑥 → 𝐵 ≼ 𝑥)
74		domfi 9115	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≼ 𝑥) → 𝐵 ∈ Fin)
75	73, 74	sylan2 593	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≺ 𝑥) → 𝐵 ∈ Fin)
76	75	3adant3 1132	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≺ 𝑥 ∧ 𝑥 ≼ 𝐴) → 𝐵 ∈ Fin)
77		sdomdomtrfi 9127	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ Fin ∧ 𝐵 ≺ 𝑥 ∧ 𝑥 ≼ 𝐴) → 𝐵 ≺ 𝐴)
78	76, 77	syld3an1 1412	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≺ 𝑥 ∧ 𝑥 ≼ 𝐴) → 𝐵 ≺ 𝐴)
79	72, 78	syl3an3 1165	. . . . . . . . 9 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≺ 𝑥 ∧ 𝑥 ≈ 𝐴) → 𝐵 ≺ 𝐴)
80	71, 79	syld3an3 1411	. . . . . . . 8 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≺ 𝑥 ∧ 𝐴 ≈ 𝑥) → 𝐵 ≺ 𝐴)
81	41, 80	syl3an1 1163	. . . . . . 7 ⊢ ((𝑥 ∈ ω ∧ 𝐵 ≺ 𝑥 ∧ 𝐴 ≈ 𝑥) → 𝐵 ≺ 𝐴)
82	81	3com23 1126	. . . . . 6 ⊢ ((𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥 ∧ 𝐵 ≺ 𝑥) → 𝐵 ≺ 𝐴)
83	82	3exp 1119	. . . . 5 ⊢ (𝑥 ∈ ω → (𝐴 ≈ 𝑥 → (𝐵 ≺ 𝑥 → 𝐵 ≺ 𝐴)))
84	68, 83	syldd 72	. . . 4 ⊢ (𝑥 ∈ ω → (𝐴 ≈ 𝑥 → (𝐵 ⊊ 𝐴 → 𝐵 ≺ 𝐴)))
85	84	rexlimiv 3130	. . 3 ⊢ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → (𝐵 ⊊ 𝐴 → 𝐵 ≺ 𝐴))
86	1, 85	sylbi 217	. 2 ⊢ (𝐴 ∈ Fin → (𝐵 ⊊ 𝐴 → 𝐵 ≺ 𝐴))
87	86	imp 406	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∃wrex 3060   ∖ cdif 3898   ⊆ wss 3901   ⊊ wpss 3902  ∅c0 4285   class class class wbr 5098  ^◡ccnv 5623  dom cdm 5624  ran crn 5625   ↾ cres 5626   “ cima 5627  Fun wfun 6486   Fn wfn 6487  –1-1→wf1 6489  –onto→wfo 6490  –1-1-onto→wf1o 6491  ‘cfv 6492  ωcom 7808   ≈ cen 8882   ≼ cdom 8883   ≺ csdm 8884  Fincfn 8885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-om 7809  df-1o 8397  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889

This theorem is referenced by:  phpeqd  9138  onomeneq  9140  pssinf  9164  f1finf1o  9175  findcard3  9185  fofinf1o  9234  ackbij1b  10150  fincssdom  10235  fin23lem25  10236  canthp1lem2  10566  pwfseqlem4  10575  uzindi  13907  symggen  19401  pgpssslw  19545  pgpfaclem2  20015  ppiltx  27145  finminlem  36514  lindsenlbs  37818