Theorem alephle 9971

Description: The argument of the aleph function is less than or equal to its value. Exercise 2 of [TakeutiZaring] p. 91. (Later, in alephfp2 9992, we will that equality can sometimes hold.) (Contributed by NM, 9-Nov-2003.) (Proof shortened by Mario Carneiro, 22-Feb-2013.)

Assertion

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

Proof of Theorem alephle

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

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
2		fveq2 6817	. . 3 ⊢ (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
3	1, 2	sseq12d 3966	. 2 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ (ℵ‘𝑥) ↔ 𝑦 ⊆ (ℵ‘𝑦)))
4		id 22	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
5		fveq2 6817	. . 3 ⊢ (𝑥 = 𝐴 → (ℵ‘𝑥) = (ℵ‘𝐴))
6	4, 5	sseq12d 3966	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ (ℵ‘𝑥) ↔ 𝐴 ⊆ (ℵ‘𝐴)))
7		alephord2i 9960	. . . . . 6 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → (ℵ‘𝑦) ∈ (ℵ‘𝑥)))
8	7	imp 406	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (ℵ‘𝑦) ∈ (ℵ‘𝑥))
9		onelon 6327	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
10		alephon 9952	. . . . . 6 ⊢ (ℵ‘𝑥) ∈ On
11		ontr2 6350	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ (ℵ‘𝑥) ∈ On) → ((𝑦 ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝑥)) → 𝑦 ∈ (ℵ‘𝑥)))
12	9, 10, 11	sylancl 586	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ((𝑦 ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝑥)) → 𝑦 ∈ (ℵ‘𝑥)))
13	8, 12	mpan2d 694	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑦 ⊆ (ℵ‘𝑦) → 𝑦 ∈ (ℵ‘𝑥)))
14	13	ralimdva 3142	. . 3 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝑦 ⊆ (ℵ‘𝑦) → ∀𝑦 ∈ 𝑥 𝑦 ∈ (ℵ‘𝑥)))
15	10	onirri 6416	. . . . 5 ⊢ ¬ (ℵ‘𝑥) ∈ (ℵ‘𝑥)
16		eleq1 2817	. . . . . 6 ⊢ (𝑦 = (ℵ‘𝑥) → (𝑦 ∈ (ℵ‘𝑥) ↔ (ℵ‘𝑥) ∈ (ℵ‘𝑥)))
17	16	rspccv 3572	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ∈ (ℵ‘𝑥) → ((ℵ‘𝑥) ∈ 𝑥 → (ℵ‘𝑥) ∈ (ℵ‘𝑥)))
18	15, 17	mtoi 199	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ∈ (ℵ‘𝑥) → ¬ (ℵ‘𝑥) ∈ 𝑥)
19		ontri1 6336	. . . . 5 ⊢ ((𝑥 ∈ On ∧ (ℵ‘𝑥) ∈ On) → (𝑥 ⊆ (ℵ‘𝑥) ↔ ¬ (ℵ‘𝑥) ∈ 𝑥))
20	10, 19	mpan2 691	. . . 4 ⊢ (𝑥 ∈ On → (𝑥 ⊆ (ℵ‘𝑥) ↔ ¬ (ℵ‘𝑥) ∈ 𝑥))
21	18, 20	imbitrrid 246	. . 3 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝑦 ∈ (ℵ‘𝑥) → 𝑥 ⊆ (ℵ‘𝑥)))
22	14, 21	syld 47	. 2 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝑦 ⊆ (ℵ‘𝑦) → 𝑥 ⊆ (ℵ‘𝑥)))
23	3, 6, 22	tfis3 7783	1 ⊢ (𝐴 ∈ On → 𝐴 ⊆ (ℵ‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2110  ∀wral 3045   ⊆ wss 3900  Oncon0 6302  ‘cfv 6477  ℵcale 9821

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 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-inf2 9526

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 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-isom 6486  df-riota 7298  df-ov 7344  df-om 7792  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-er 8617  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-oi 9391  df-har 9438  df-card 9824  df-aleph 9825

This theorem is referenced by:  cardaleph  9972  alephfp  9991  winafp  10580