Theorem php3 9137

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 5303. (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 8916	. . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
2		bren 8897	. . . . . 6 ⊢ (𝐴 ≈ 𝑥 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝑥)
3		pssss 4039	. . . . . . . . . . . . . 14 ⊢ (𝐵 ⊊ 𝐴 → 𝐵 ⊆ 𝐴)
4		imass2 6062	. . . . . . . . . . . . . 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 4412	. . . . . . . . . . . . . 14 ⊢ (𝐵 ⊊ 𝐴 → ∃𝑦(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))
8		eldif 3900	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (𝐴 ∖ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))
9		f1ofn 6776	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → 𝑓 Fn 𝐴)
10		difss 4077	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
11		fnfvima 7182	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 Fn 𝐴 ∧ (𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ 𝑦 ∈ (𝐴 ∖ 𝐵)) → (𝑓‘𝑦) ∈ (𝑓 “ (𝐴 ∖ 𝐵)))
12	11	3expia 1122	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 Fn 𝐴 ∧ (𝐴 ∖ 𝐵) ⊆ 𝐴) → (𝑦 ∈ (𝐴 ∖ 𝐵) → (𝑓‘𝑦) ∈ (𝑓 “ (𝐴 ∖ 𝐵))))
13	9, 10, 12	sylancl 587	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑦 ∈ (𝐴 ∖ 𝐵) → (𝑓‘𝑦) ∈ (𝑓 “ (𝐴 ∖ 𝐵))))
14		dff1o3 6781	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓:𝐴–1-1-onto→𝑥 ↔ (𝑓:𝐴–onto→𝑥 ∧ Fun ◡𝑓))
15		imadif 6577	. . . . . . . . . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . . . . . . 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 4281	. . . . . . . . . . . . . . . . . 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 1935	. . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → ¬ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅)
25		ssdif0 4307	. . . . . . . . . . . . 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 4030	. . . . . . . . . . . 12 ⊢ ((𝑓 “ 𝐵) ⊊ (𝑓 “ 𝐴) ↔ ((𝑓 “ 𝐵) ⊆ (𝑓 “ 𝐴) ∧ ¬ (𝑓 “ 𝐴) ⊆ (𝑓 “ 𝐵)))
28	6, 26, 27	sylanbrc 584	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → (𝑓 “ 𝐵) ⊊ (𝑓 “ 𝐴))
29		imadmrn 6030	. . . . . . . . . . . . . 14 ⊢ (𝑓 “ dom 𝑓) = ran 𝑓
30		f1odm 6779	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → dom 𝑓 = 𝐴)
31	30	imaeq2d 6020	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑓 “ dom 𝑓) = (𝑓 “ 𝐴))
32		f1ofo 6782	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → 𝑓:𝐴–onto→𝑥)
33		forn 6750	. . . . . . . . . . . . . . 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 2796	. . . . . . . . . . . . 13 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑓 “ 𝐴) = 𝑥)
36	35	psseq2d 4037	. . . . . . . . . . . 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 9136	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ω ∧ (𝑓 “ 𝐵) ⊊ 𝑥) → (𝑓 “ 𝐵) ≺ 𝑥)
40	38, 39	sylan2 594	. . . . . . . . 9 ⊢ ((𝑥 ∈ ω ∧ (𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴)) → (𝑓 “ 𝐵) ≺ 𝑥)
41		nnfi 9096	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω → 𝑥 ∈ Fin)
42		f1of1 6774	. . . . . . . . . . . 12 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → 𝑓:𝐴–1-1→𝑥)
43		f1ores 6789	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐴–1-1→𝑥 ∧ 𝐵 ⊆ 𝐴) → (𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵))
44	42, 3, 43	syl2an 597	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → (𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵))
45		vex 3434	. . . . . . . . . . . . . 14 ⊢ 𝑓 ∈ V
46	45	resex 5989	. . . . . . . . . . . . 13 ⊢ (𝑓 ↾ 𝐵) ∈ V
47		f1oeq1 6763	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑓 ↾ 𝐵) → (𝑦:𝐵–1-1-onto→(𝑓 “ 𝐵) ↔ (𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵)))
48	46, 47	spcev 3549	. . . . . . . . . . . 12 ⊢ ((𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵) → ∃𝑦 𝑦:𝐵–1-1-onto→(𝑓 “ 𝐵))
49		bren 8897	. . . . . . . . . . . 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 8920	. . . . . . . . . . . 12 ⊢ (𝐵 ≈ (𝑓 “ 𝐵) → 𝐵 ≼ (𝑓 “ 𝐵))
53		sdomdom 8921	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 “ 𝐵) ≺ 𝑥 → (𝑓 “ 𝐵) ≼ 𝑥)
54		domfi 9117	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Fin ∧ (𝑓 “ 𝐵) ≼ 𝑥) → (𝑓 “ 𝐵) ∈ Fin)
55	53, 54	sylan2 594	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Fin ∧ (𝑓 “ 𝐵) ≺ 𝑥) → (𝑓 “ 𝐵) ∈ Fin)
56	55	3adant2 1132	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≼ (𝑓 “ 𝐵) ∧ (𝑓 “ 𝐵) ≺ 𝑥) → (𝑓 “ 𝐵) ∈ Fin)
57		domfi 9117	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 “ 𝐵) ∈ Fin ∧ 𝐵 ≼ (𝑓 “ 𝐵)) → 𝐵 ∈ Fin)
58	57	3adant3 1133	. . . . . . . . . . . . . 14 ⊢ (((𝑓 “ 𝐵) ∈ Fin ∧ 𝐵 ≼ (𝑓 “ 𝐵) ∧ (𝑓 “ 𝐵) ≺ 𝑥) → 𝐵 ∈ Fin)
59		domsdomtrfi 9130	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ Fin ∧ 𝐵 ≼ (𝑓 “ 𝐵) ∧ (𝑓 “ 𝐵) ≺ 𝑥) → 𝐵 ≺ 𝑥)
60	58, 59	syld3an1 1413	. . . . . . . . . . . . 13 ⊢ (((𝑓 “ 𝐵) ∈ Fin ∧ 𝐵 ≼ (𝑓 “ 𝐵) ∧ (𝑓 “ 𝐵) ≺ 𝑥) → 𝐵 ≺ 𝑥)
61	56, 60	syld3an1 1413	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≼ (𝑓 “ 𝐵) ∧ (𝑓 “ 𝐵) ≺ 𝑥) → 𝐵 ≺ 𝑥)
62	52, 61	syl3an2 1165	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≈ (𝑓 “ 𝐵) ∧ (𝑓 “ 𝐵) ≺ 𝑥) → 𝐵 ≺ 𝑥)
63	62	3expia 1122	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≈ (𝑓 “ 𝐵)) → ((𝑓 “ 𝐵) ≺ 𝑥 → 𝐵 ≺ 𝑥))
64	41, 51, 63	syl2an 597	. . . . . . . . 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 1935	. . . . . 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 9112	. . . . . . . . . . 11 ⊢ (𝑥 ∈ Fin → (𝑥 ≈ 𝐴 ↔ 𝐴 ≈ 𝑥))
70	69	adantr 480	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≺ 𝑥) → (𝑥 ≈ 𝐴 ↔ 𝐴 ≈ 𝑥))
71	70	biimp3ar 1473	. . . . . . . . 9 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≺ 𝑥 ∧ 𝐴 ≈ 𝑥) → 𝑥 ≈ 𝐴)
72		endom 8920	. . . . . . . . . 10 ⊢ (𝑥 ≈ 𝐴 → 𝑥 ≼ 𝐴)
73		sdomdom 8921	. . . . . . . . . . . . 13 ⊢ (𝐵 ≺ 𝑥 → 𝐵 ≼ 𝑥)
74		domfi 9117	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≼ 𝑥) → 𝐵 ∈ Fin)
75	73, 74	sylan2 594	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≺ 𝑥) → 𝐵 ∈ Fin)
76	75	3adant3 1133	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≺ 𝑥 ∧ 𝑥 ≼ 𝐴) → 𝐵 ∈ Fin)
77		sdomdomtrfi 9129	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ Fin ∧ 𝐵 ≺ 𝑥 ∧ 𝑥 ≼ 𝐴) → 𝐵 ≺ 𝐴)
78	76, 77	syld3an1 1413	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≺ 𝑥 ∧ 𝑥 ≼ 𝐴) → 𝐵 ≺ 𝐴)
79	72, 78	syl3an3 1166	. . . . . . . . 9 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≺ 𝑥 ∧ 𝑥 ≈ 𝐴) → 𝐵 ≺ 𝐴)
80	71, 79	syld3an3 1412	. . . . . . . 8 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≺ 𝑥 ∧ 𝐴 ≈ 𝑥) → 𝐵 ≺ 𝐴)
81	41, 80	syl3an1 1164	. . . . . . 7 ⊢ ((𝑥 ∈ ω ∧ 𝐵 ≺ 𝑥 ∧ 𝐴 ≈ 𝑥) → 𝐵 ≺ 𝐴)
82	81	3com23 1127	. . . . . 6 ⊢ ((𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥 ∧ 𝐵 ≺ 𝑥) → 𝐵 ≺ 𝐴)
83	82	3exp 1120	. . . . 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 3132	. . 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 1542  ∃wex 1781   ∈ wcel 2114  ∃wrex 3062   ∖ cdif 3887   ⊆ wss 3890   ⊊ wpss 3891  ∅c0 4274   class class class wbr 5086  ^◡ccnv 5624  dom cdm 5625  ran crn 5626   ↾ cres 5627   “ cima 5628  Fun wfun 6487   Fn wfn 6488  –1-1→wf1 6490  –onto→wfo 6491  –1-1-onto→wf1o 6492  ‘cfv 6493  ωcom 7811   ≈ cen 8884   ≼ cdom 8885   ≺ csdm 8886  Fincfn 8887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-om 7812  df-1o 8399  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891

This theorem is referenced by:  phpeqd  9140  onomeneq  9142  pssinf  9166  f1finf1o  9177  findcard3  9187  fofinf1o  9236  ackbij1b  10154  fincssdom  10239  fin23lem25  10240  canthp1lem2  10570  pwfseqlem4  10579  uzindi  13938  symggen  19439  pgpssslw  19583  pgpfaclem2  20053  ppiltx  27157  finminlem  36519  lindsenlbs  37953