Theorem php3 9134

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 5295. (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 8913	. . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
2		bren 8894	. . . . . 6 ⊢ (𝐴 ≈ 𝑥 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝑥)
3		pssss 4030	. . . . . . . . . . . . . 14 ⊢ (𝐵 ⊊ 𝐴 → 𝐵 ⊆ 𝐴)
4		imass2 6055	. . . . . . . . . . . . . 14 ⊢ (𝐵 ⊆ 𝐴 → (𝑓 “ 𝐵) ⊆ (𝑓 “ 𝐴))
5	3, 4	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐵 ⊊ 𝐴 → (𝑓 “ 𝐵) ⊆ (𝑓 “ 𝐴))
6	5	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → (𝑓 “ 𝐵) ⊆ (𝑓 “ 𝐴))
7		pssnel 4400	. . . . . . . . . . . . . 14 ⊢ (𝐵 ⊊ 𝐴 → ∃𝑦(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))
8		eldif 3893	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (𝐴 ∖ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))
9		f1ofn 6769	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → 𝑓 Fn 𝐴)
10		difss 4067	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
11		fnfvima 7178	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 Fn 𝐴 ∧ (𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ 𝑦 ∈ (𝐴 ∖ 𝐵)) → (𝑓‘𝑦) ∈ (𝑓 “ (𝐴 ∖ 𝐵)))
12	11	3expia 1127	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 Fn 𝐴 ∧ (𝐴 ∖ 𝐵) ⊆ 𝐴) → (𝑦 ∈ (𝐴 ∖ 𝐵) → (𝑓‘𝑦) ∈ (𝑓 “ (𝐴 ∖ 𝐵))))
13	9, 10, 12	sylancl 592	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑦 ∈ (𝐴 ∖ 𝐵) → (𝑓‘𝑦) ∈ (𝑓 “ (𝐴 ∖ 𝐵))))
14		dff1o3 6774	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓:𝐴–1-1-onto→𝑥 ↔ (𝑓:𝐴–onto→𝑥 ∧ Fun ◡𝑓))
15		imadif 6570	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Fun ◡𝑓 → (𝑓 “ (𝐴 ∖ 𝐵)) = ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)))
16	14, 15	simplbiim 509	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑓 “ (𝐴 ∖ 𝐵)) = ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)))
17	16	eleq2d 2825	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → ((𝑓‘𝑦) ∈ (𝑓 “ (𝐴 ∖ 𝐵)) ↔ (𝑓‘𝑦) ∈ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵))))
18	13, 17	sylibd 240	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑦 ∈ (𝐴 ∖ 𝐵) → (𝑓‘𝑦) ∈ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵))))
19		n0i 4269	. . . . . . . . . . . . . . . . . 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 244	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵) → ¬ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅))
22	21	exlimdv 1940	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (∃𝑦(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵) → ¬ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅))
23	22	imp 407	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ ∃𝑦(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)) → ¬ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅)
24	7, 23	sylan2 599	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → ¬ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅)
25		ssdif0 4295	. . . . . . . . . . . . 13 ⊢ ((𝑓 “ 𝐴) ⊆ (𝑓 “ 𝐵) ↔ ((𝑓 “ 𝐴) ∖ (𝑓 “ 𝐵)) = ∅)
26	24, 25	sylnibr 330	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → ¬ (𝑓 “ 𝐴) ⊆ (𝑓 “ 𝐵))
27		dfpss3 4021	. . . . . . . . . . . 12 ⊢ ((𝑓 “ 𝐵) ⊊ (𝑓 “ 𝐴) ↔ ((𝑓 “ 𝐵) ⊆ (𝑓 “ 𝐴) ∧ ¬ (𝑓 “ 𝐴) ⊆ (𝑓 “ 𝐵)))
28	6, 26, 27	sylanbrc 589	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → (𝑓 “ 𝐵) ⊊ (𝑓 “ 𝐴))
29		imadmrn 6023	. . . . . . . . . . . . . 14 ⊢ (𝑓 “ dom 𝑓) = ran 𝑓
30		f1odm 6772	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → dom 𝑓 = 𝐴)
31	30	imaeq2d 6013	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑓 “ dom 𝑓) = (𝑓 “ 𝐴))
32		f1ofo 6775	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → 𝑓:𝐴–onto→𝑥)
33		forn 6743	. . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . 13 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → (𝑓 “ 𝐴) = 𝑥)
36	35	psseq2d 4028	. . . . . . . . . . . 12 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → ((𝑓 “ 𝐵) ⊊ (𝑓 “ 𝐴) ↔ (𝑓 “ 𝐵) ⊊ 𝑥))
37	36	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → ((𝑓 “ 𝐵) ⊊ (𝑓 “ 𝐴) ↔ (𝑓 “ 𝐵) ⊊ 𝑥))
38	28, 37	mpbid 233	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → (𝑓 “ 𝐵) ⊊ 𝑥)
39		php2 9133	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ω ∧ (𝑓 “ 𝐵) ⊊ 𝑥) → (𝑓 “ 𝐵) ≺ 𝑥)
40	38, 39	sylan2 599	. . . . . . . . 9 ⊢ ((𝑥 ∈ ω ∧ (𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴)) → (𝑓 “ 𝐵) ≺ 𝑥)
41		nnfi 9093	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω → 𝑥 ∈ Fin)
42		f1of1 6767	. . . . . . . . . . . 12 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → 𝑓:𝐴–1-1→𝑥)
43		f1ores 6782	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐴–1-1→𝑥 ∧ 𝐵 ⊆ 𝐴) → (𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵))
44	42, 3, 43	syl2an 602	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊊ 𝐴) → (𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵))
45		vex 3435	. . . . . . . . . . . . . 14 ⊢ 𝑓 ∈ V
46	45	resex 5982	. . . . . . . . . . . . 13 ⊢ (𝑓 ↾ 𝐵) ∈ V
47		f1oeq1 6756	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑓 ↾ 𝐵) → (𝑦:𝐵–1-1-onto→(𝑓 “ 𝐵) ↔ (𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵)))
48	46, 47	spcev 3544	. . . . . . . . . . . 12 ⊢ ((𝑓 ↾ 𝐵):𝐵–1-1-onto→(𝑓 “ 𝐵) → ∃𝑦 𝑦:𝐵–1-1-onto→(𝑓 “ 𝐵))
49		bren 8894	. . . . . . . . . . . 12 ⊢ (𝐵 ≈ (𝑓 “ 𝐵) ↔ ∃𝑦 𝑦:𝐵–1-1-onto→(𝑓 “ 𝐵))
50	48, 49	sylibr 235	. . . . . . . . . . 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 8917	. . . . . . . . . . . 12 ⊢ (𝐵 ≈ (𝑓 “ 𝐵) → 𝐵 ≼ (𝑓 “ 𝐵))
53		sdomdom 8918	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 “ 𝐵) ≺ 𝑥 → (𝑓 “ 𝐵) ≼ 𝑥)
54		domfi 9114	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Fin ∧ (𝑓 “ 𝐵) ≼ 𝑥) → (𝑓 “ 𝐵) ∈ Fin)
55	53, 54	sylan2 599	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Fin ∧ (𝑓 “ 𝐵) ≺ 𝑥) → (𝑓 “ 𝐵) ∈ Fin)
56	55	3adant2 1137	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≼ (𝑓 “ 𝐵) ∧ (𝑓 “ 𝐵) ≺ 𝑥) → (𝑓 “ 𝐵) ∈ Fin)
57		domfi 9114	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 “ 𝐵) ∈ Fin ∧ 𝐵 ≼ (𝑓 “ 𝐵)) → 𝐵 ∈ Fin)
58	57	3adant3 1138	. . . . . . . . . . . . . 14 ⊢ (((𝑓 “ 𝐵) ∈ Fin ∧ 𝐵 ≼ (𝑓 “ 𝐵) ∧ (𝑓 “ 𝐵) ≺ 𝑥) → 𝐵 ∈ Fin)
59		domsdomtrfi 9127	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ Fin ∧ 𝐵 ≼ (𝑓 “ 𝐵) ∧ (𝑓 “ 𝐵) ≺ 𝑥) → 𝐵 ≺ 𝑥)
60	58, 59	syld3an1 1418	. . . . . . . . . . . . 13 ⊢ (((𝑓 “ 𝐵) ∈ Fin ∧ 𝐵 ≼ (𝑓 “ 𝐵) ∧ (𝑓 “ 𝐵) ≺ 𝑥) → 𝐵 ≺ 𝑥)
61	56, 60	syld3an1 1418	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≼ (𝑓 “ 𝐵) ∧ (𝑓 “ 𝐵) ≺ 𝑥) → 𝐵 ≺ 𝑥)
62	52, 61	syl3an2 1170	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≈ (𝑓 “ 𝐵) ∧ (𝑓 “ 𝐵) ≺ 𝑥) → 𝐵 ≺ 𝑥)
63	62	3expia 1127	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≈ (𝑓 “ 𝐵)) → ((𝑓 “ 𝐵) ≺ 𝑥 → 𝐵 ≺ 𝑥))
64	41, 51, 63	syl2an 602	. . . . . . . . 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 421	. . . . . . 7 ⊢ (𝑥 ∈ ω → (𝑓:𝐴–1-1-onto→𝑥 → (𝐵 ⊊ 𝐴 → 𝐵 ≺ 𝑥)))
67	66	exlimdv 1940	. . . . . 6 ⊢ (𝑥 ∈ ω → (∃𝑓 𝑓:𝐴–1-1-onto→𝑥 → (𝐵 ⊊ 𝐴 → 𝐵 ≺ 𝑥)))
68	2, 67	biimtrid 243	. . . . 5 ⊢ (𝑥 ∈ ω → (𝐴 ≈ 𝑥 → (𝐵 ⊊ 𝐴 → 𝐵 ≺ 𝑥)))
69		ensymfib 9109	. . . . . . . . . . 11 ⊢ (𝑥 ∈ Fin → (𝑥 ≈ 𝐴 ↔ 𝐴 ≈ 𝑥))
70	69	adantr 481	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≺ 𝑥) → (𝑥 ≈ 𝐴 ↔ 𝐴 ≈ 𝑥))
71	70	biimp3ar 1478	. . . . . . . . 9 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≺ 𝑥 ∧ 𝐴 ≈ 𝑥) → 𝑥 ≈ 𝐴)
72		endom 8917	. . . . . . . . . 10 ⊢ (𝑥 ≈ 𝐴 → 𝑥 ≼ 𝐴)
73		sdomdom 8918	. . . . . . . . . . . . 13 ⊢ (𝐵 ≺ 𝑥 → 𝐵 ≼ 𝑥)
74		domfi 9114	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≼ 𝑥) → 𝐵 ∈ Fin)
75	73, 74	sylan2 599	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≺ 𝑥) → 𝐵 ∈ Fin)
76	75	3adant3 1138	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≺ 𝑥 ∧ 𝑥 ≼ 𝐴) → 𝐵 ∈ Fin)
77		sdomdomtrfi 9126	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ Fin ∧ 𝐵 ≺ 𝑥 ∧ 𝑥 ≼ 𝐴) → 𝐵 ≺ 𝐴)
78	76, 77	syld3an1 1418	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≺ 𝑥 ∧ 𝑥 ≼ 𝐴) → 𝐵 ≺ 𝐴)
79	72, 78	syl3an3 1171	. . . . . . . . 9 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≺ 𝑥 ∧ 𝑥 ≈ 𝐴) → 𝐵 ≺ 𝐴)
80	71, 79	syld3an3 1417	. . . . . . . 8 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≺ 𝑥 ∧ 𝐴 ≈ 𝑥) → 𝐵 ≺ 𝐴)
81	41, 80	syl3an1 1169	. . . . . . 7 ⊢ ((𝑥 ∈ ω ∧ 𝐵 ≺ 𝑥 ∧ 𝐴 ≈ 𝑥) → 𝐵 ≺ 𝐴)
82	81	3com23 1132	. . . . . 6 ⊢ ((𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥 ∧ 𝐵 ≺ 𝑥) → 𝐵 ≺ 𝐴)
83	82	3exp 1125	. . . . 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 3133	. . 3 ⊢ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → (𝐵 ⊊ 𝐴 → 𝐵 ≺ 𝐴))
86	1, 85	sylbi 218	. 2 ⊢ (𝐴 ∈ Fin → (𝐵 ⊊ 𝐴 → 𝐵 ≺ 𝐴))
87	86	imp 407	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ∃wrex 3063   ∖ cdif 3880   ⊆ wss 3883   ⊊ wpss 3884  ∅c0 4262   class class class wbr 5073  ^◡ccnv 5618  dom cdm 5619  ran crn 5620   ↾ cres 5621   “ cima 5622  Fun wfun 6480   Fn wfn 6481  –1-1→wf1 6483  –onto→wfo 6484  –1-1-onto→wf1o 6485  ‘cfv 6486  ωcom 7807   ≈ cen 8881   ≼ cdom 8882   ≺ csdm 8883  Fincfn 8884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5219  ax-nul 5229  ax-pr 5363  ax-un 7679

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-br 5074  df-opab 5136  df-mpt 5155  df-tr 5181  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-om 7808  df-1o 8396  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888

This theorem is referenced by:  phpeqd  9137  onomeneq  9139  pssinf  9163  f1finf1o  9174  findcard3  9184  fofinf1o  9233  ackbij1b  10152  fincssdom  10237  fin23lem25  10238  canthp1lem2  10568  pwfseqlem4  10577  uzindi  13936  symggen  19437  pgpssslw  19581  pgpfaclem2  20051  ppiltx  27159  finminlem  36555  lindsenlbs  37991