Theorem php3 9275

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 5383. (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 9036	. . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
2		bren 9013	. . . . . 6 ⊢ (𝐴 ≈ 𝑥 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝑥)
3		pssss 4121	. . . . . . . . . . . . . 14 ⊢ (𝐵 ⊊ 𝐴 → 𝐵 ⊆ 𝐴)
4		imass2 6132	. . . . . . . . . . . . . 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 4494	. . . . . . . . . . . . . 14 ⊢ (𝐵 ⊊ 𝐴 → ∃𝑦(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))
8		eldif 3986	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (𝐴 ∖ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))
9		f1ofn 6863	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → 𝑓 Fn 𝐴)
10		difss 4159	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
11		fnfvima 7270	. . . . . . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑦 ∈ (𝐴 ∖ 𝐵) → (𝑓‘𝑦) ∈ (𝑓 “ (𝐴 ∖ 𝐵))))
14		dff1o3 6868	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓:𝐴–1-1-onto→𝑥 ↔ (𝑓:𝐴–onto→𝑥 ∧ Fun ◡𝑓))
15		imadif 6662	. . . . . . . . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . . . . . . . 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 4363	. . . . . . . . . . . . . . . . . 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 1932	. . . . . . . . . . . . . . 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 592	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → ¬ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅)
25		ssdif0 4389	. . . . . . . . . . . . 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 4112	. . . . . . . . . . . 12 ⊢ ((𝑓 “ 𝐵) ⊊ (𝑓 “ 𝐴) ↔ ((𝑓 “ 𝐵) ⊆ (𝑓 “ 𝐴) ∧ ¬ (𝑓 “ 𝐴) ⊆ (𝑓 “ 𝐵)))
28	6, 26, 27	sylanbrc 582	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → (𝑓 “ 𝐵) ⊊ (𝑓 “ 𝐴))
29		imadmrn 6099	. . . . . . . . . . . . . 14 ⊢ (𝑓 “ dom 𝑓) = ran 𝑓
30		f1odm 6866	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → dom 𝑓 = 𝐴)
31	30	imaeq2d 6089	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑓 “ dom 𝑓) = (𝑓 “ 𝐴))
32		f1ofo 6869	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → 𝑓:𝐴–onto→𝑥)
33		forn 6837	. . . . . . . . . . . . . . 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 2804	. . . . . . . . . . . . 13 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑓 “ 𝐴) = 𝑥)
36	35	psseq2d 4119	. . . . . . . . . . . 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 9274	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ω ∧ (𝑓 “ 𝐵) ⊊ 𝑥) → (𝑓 “ 𝐵) ≺ 𝑥)
40	38, 39	sylan2 592	. . . . . . . . 9 ⊢ ((𝑥 ∈ ω ∧ (𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴)) → (𝑓 “ 𝐵) ≺ 𝑥)
41		nnfi 9233	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω → 𝑥 ∈ Fin)
42		f1of1 6861	. . . . . . . . . . . 12 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → 𝑓:𝐴–1-1→𝑥)
43		f1ores 6876	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐴–1-1→𝑥 ∧ 𝐵 ⊆ 𝐴) → (𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵))
44	42, 3, 43	syl2an 595	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → (𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵))
45		vex 3492	. . . . . . . . . . . . . 14 ⊢ 𝑓 ∈ V
46	45	resex 6058	. . . . . . . . . . . . 13 ⊢ (𝑓 ↾ 𝐵) ∈ V
47		f1oeq1 6850	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑓 ↾ 𝐵) → (𝑦:𝐵–1-1-onto→(𝑓 “ 𝐵) ↔ (𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵)))
48	46, 47	spcev 3619	. . . . . . . . . . . 12 ⊢ ((𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵) → ∃𝑦 𝑦:𝐵–1-1-onto→(𝑓 “ 𝐵))
49		bren 9013	. . . . . . . . . . . 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 9039	. . . . . . . . . . . 12 ⊢ (𝐵 ≈ (𝑓 “ 𝐵) → 𝐵 ≼ (𝑓 “ 𝐵))
53		sdomdom 9040	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 “ 𝐵) ≺ 𝑥 → (𝑓 “ 𝐵) ≼ 𝑥)
54		domfi 9255	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Fin ∧ (𝑓 “ 𝐵) ≼ 𝑥) → (𝑓 “ 𝐵) ∈ Fin)
55	53, 54	sylan2 592	. . . . . . . . . . . . . 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 9255	. . . . . . . . . . . . . . 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 9268	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ Fin ∧ 𝐵 ≼ (𝑓 “ 𝐵) ∧ (𝑓 “ 𝐵) ≺ 𝑥) → 𝐵 ≺ 𝑥)
60	58, 59	syld3an1 1410	. . . . . . . . . . . . 13 ⊢ (((𝑓 “ 𝐵) ∈ Fin ∧ 𝐵 ≼ (𝑓 “ 𝐵) ∧ (𝑓 “ 𝐵) ≺ 𝑥) → 𝐵 ≺ 𝑥)
61	56, 60	syld3an1 1410	. . . . . . . . . . . 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 595	. . . . . . . . 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 1932	. . . . . 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 9250	. . . . . . . . . . 11 ⊢ (𝑥 ∈ Fin → (𝑥 ≈ 𝐴 ↔ 𝐴 ≈ 𝑥))
70	69	adantr 480	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≺ 𝑥) → (𝑥 ≈ 𝐴 ↔ 𝐴 ≈ 𝑥))
71	70	biimp3ar 1470	. . . . . . . . 9 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≺ 𝑥 ∧ 𝐴 ≈ 𝑥) → 𝑥 ≈ 𝐴)
72		endom 9039	. . . . . . . . . 10 ⊢ (𝑥 ≈ 𝐴 → 𝑥 ≼ 𝐴)
73		sdomdom 9040	. . . . . . . . . . . . 13 ⊢ (𝐵 ≺ 𝑥 → 𝐵 ≼ 𝑥)
74		domfi 9255	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≼ 𝑥) → 𝐵 ∈ Fin)
75	73, 74	sylan2 592	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≺ 𝑥) → 𝐵 ∈ Fin)
76	75	3adant3 1132	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≺ 𝑥 ∧ 𝑥 ≼ 𝐴) → 𝐵 ∈ Fin)
77		sdomdomtrfi 9267	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ Fin ∧ 𝐵 ≺ 𝑥 ∧ 𝑥 ≼ 𝐴) → 𝐵 ≺ 𝐴)
78	76, 77	syld3an1 1410	. . . . . . . . . 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 1409	. . . . . . . 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 3154	. . 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 1537  ∃wex 1777   ∈ wcel 2108  ∃wrex 3076   ∖ cdif 3973   ⊆ wss 3976   ⊊ wpss 3977  ∅c0 4352   class class class wbr 5166  ^◡ccnv 5699  dom cdm 5700  ran crn 5701   ↾ cres 5702   “ cima 5703  Fun wfun 6567   Fn wfn 6568  –1-1→wf1 6570  –onto→wfo 6571  –1-1-onto→wf1o 6572  ‘cfv 6573  ωcom 7903   ≈ cen 9000   ≼ cdom 9001   ≺ csdm 9002  Fincfn 9003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-om 7904  df-1o 8522  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007

This theorem is referenced by:  phpeqd  9278  phpeqdOLD  9288  onomeneq  9291  pssinf  9319  f1finf1o  9333  f1finf1oOLD  9334  findcard3  9346  findcard3OLD  9347  fofinf1o  9400  ackbij1b  10307  fincssdom  10392  fin23lem25  10393  canthp1lem2  10722  pwfseqlem4  10731  uzindi  14033  symggen  19512  pgpssslw  19656  pgpfaclem2  20126  ppiltx  27238  finminlem  36284  lindsenlbs  37575