Theorem alephle 3689

Description: The argument of the aleph function is less than or equal to its value. Exercise 2 of [TakeutiZaring] p. 91. (Later, in alephfp2 (future), we will that equality can sometimes hold.)

Assertion

Ref	Expression
alephle	⊢ (A ∈ On → A ⊆ (ℵ ‘A))

Proof of Theorem alephle

Step	Hyp	Ref	Expression
1		id 9	. . 3 ⊢ (x = y → x = y)
2		fveq2 2832	. . 3 ⊢ (x = y → (ℵ ‘x) = (ℵ ‘y))
3	1, 2	sseq12d 1529	. 2 ⊢ (x = y → (x ⊆ (ℵ ‘x) ↔ y ⊆ (ℵ ‘y)))
4		id 9	. . 3 ⊢ (x = A → x = A)
5		fveq2 2832	. . 3 ⊢ (x = A → (ℵ ‘x) = (ℵ ‘A))
6	4, 5	sseq12d 1529	. 2 ⊢ (x = A → (x ⊆ (ℵ ‘x) ↔ A ⊆ (ℵ ‘A)))
7		alephord2i 3682	. . . . . 6 ⊢ (x ∈ On → (y ∈ x → (ℵ ‘y) ∈ (ℵ ‘x)))
8	7	imp 277	. . . . 5 ⊢ ((x ∈ On ∧ y ∈ x) → (ℵ ‘y) ∈ (ℵ ‘x))
9		onelon 2223	. . . . . 6 ⊢ ((x ∈ On ∧ y ∈ x) → y ∈ On)
10		alephon 3671	. . . . . . 7 ⊢ (ℵ ‘x) ∈ On
11		ontr2 2259	. . . . . . 7 ⊢ ((y ∈ On ∧ (ℵ ‘x) ∈ On) → ((y ⊆ (ℵ ‘y) ∧ (ℵ ‘y) ∈ (ℵ ‘x)) → y ∈ (ℵ ‘x)))
12	10, 11	mpan2 519	. . . . . 6 ⊢ (y ∈ On → ((y ⊆ (ℵ ‘y) ∧ (ℵ ‘y) ∈ (ℵ ‘x)) → y ∈ (ℵ ‘x)))
13	9, 12	syl 12	. . . . 5 ⊢ ((x ∈ On ∧ y ∈ x) → ((y ⊆ (ℵ ‘y) ∧ (ℵ ‘y) ∈ (ℵ ‘x)) → y ∈ (ℵ ‘x)))
14	8, 13	mpan2d 525	. . . 4 ⊢ ((x ∈ On ∧ y ∈ x) → (y ⊆ (ℵ ‘y) → y ∈ (ℵ ‘x)))
15	14	r19.20dva 1256	. . 3 ⊢ (x ∈ On → (∀y ∈ x y ⊆ (ℵ ‘y) → ∀y ∈ x y ∈ (ℵ ‘x)))
16		ontri1 2232	. . . . 5 ⊢ ((x ∈ On ∧ (ℵ ‘x) ∈ On) → (x ⊆ (ℵ ‘x) ↔ ¬ (ℵ ‘x) ∈ x))
17	10, 16	mpan2 519	. . . 4 ⊢ (x ∈ On → (x ⊆ (ℵ ‘x) ↔ ¬ (ℵ ‘x) ∈ x))
18	10	oneirr 2345	. . . . 5 ⊢ ¬ (ℵ ‘x) ∈ (ℵ ‘x)
19		eleq1 1149	. . . . . 6 ⊢ (y = (ℵ ‘x) → (y ∈ (ℵ ‘x) ↔ (ℵ ‘x) ∈ (ℵ ‘x)))
20	19	rcla4v 1402	. . . . 5 ⊢ (∀y ∈ x y ∈ (ℵ ‘x) → ((ℵ ‘x) ∈ x → (ℵ ‘x) ∈ (ℵ ‘x)))
21	18, 20	mtoi 94	. . . 4 ⊢ (∀y ∈ x y ∈ (ℵ ‘x) → ¬ (ℵ ‘x) ∈ x)
22	17, 21	syl5bir 184	. . 3 ⊢ (x ∈ On → (∀y ∈ x y ∈ (ℵ ‘x) → x ⊆ (ℵ ‘x)))
23	15, 22	syld 27	. 2 ⊢ (x ∈ On → (∀y ∈ x y ⊆ (ℵ ‘y) → x ⊆ (ℵ ‘x)))
24	3, 6, 23	tfis3 2248	1 ⊢ (A ∈ On → A ⊆ (ℵ ‘A))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   = weq 797   ∈ wel 803   = wceq 1091   ∈ wcel 1092  ∀wral 1201   ⊆ wss 1487  Oncon0 2199   ‘cfv 2422  ℵcale 3621

This theorem is referenced by:  cardaleph 3690

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-er 3200  df-en 3274  df-dom 3275  df-sdom 3276  df-card 3623  df-aleph 3624

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