Theorem alephle 9516

Description: The argument of the aleph function is less than or equal to its value. Exercise 2 of [TakeutiZaring] p. 91. (Later, in alephfp2 9537, 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 6672	. . 3 ⊢ (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
3	1, 2	sseq12d 4002	. 2 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ (ℵ‘𝑥) ↔ 𝑦 ⊆ (ℵ‘𝑦)))
4		id 22	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
5		fveq2 6672	. . 3 ⊢ (𝑥 = 𝐴 → (ℵ‘𝑥) = (ℵ‘𝐴))
6	4, 5	sseq12d 4002	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ (ℵ‘𝑥) ↔ 𝐴 ⊆ (ℵ‘𝐴)))
7		alephord2i 9505	. . . . . 6 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → (ℵ‘𝑦) ∈ (ℵ‘𝑥)))
8	7	imp 409	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (ℵ‘𝑦) ∈ (ℵ‘𝑥))
9		onelon 6218	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
10		alephon 9497	. . . . . 6 ⊢ (ℵ‘𝑥) ∈ On
11		ontr2 6240	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ (ℵ‘𝑥) ∈ On) → ((𝑦 ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝑥)) → 𝑦 ∈ (ℵ‘𝑥)))
12	9, 10, 11	sylancl 588	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ((𝑦 ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝑥)) → 𝑦 ∈ (ℵ‘𝑥)))
13	8, 12	mpan2d 692	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑦 ⊆ (ℵ‘𝑦) → 𝑦 ∈ (ℵ‘𝑥)))
14	13	ralimdva 3179	. . 3 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝑦 ⊆ (ℵ‘𝑦) → ∀𝑦 ∈ 𝑥 𝑦 ∈ (ℵ‘𝑥)))
15	10	onirri 6299	. . . . 5 ⊢ ¬ (ℵ‘𝑥) ∈ (ℵ‘𝑥)
16		eleq1 2902	. . . . . 6 ⊢ (𝑦 = (ℵ‘𝑥) → (𝑦 ∈ (ℵ‘𝑥) ↔ (ℵ‘𝑥) ∈ (ℵ‘𝑥)))
17	16	rspccv 3622	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ∈ (ℵ‘𝑥) → ((ℵ‘𝑥) ∈ 𝑥 → (ℵ‘𝑥) ∈ (ℵ‘𝑥)))
18	15, 17	mtoi 201	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ∈ (ℵ‘𝑥) → ¬ (ℵ‘𝑥) ∈ 𝑥)
19		ontri1 6227	. . . . 5 ⊢ ((𝑥 ∈ On ∧ (ℵ‘𝑥) ∈ On) → (𝑥 ⊆ (ℵ‘𝑥) ↔ ¬ (ℵ‘𝑥) ∈ 𝑥))
20	10, 19	mpan2 689	. . . 4 ⊢ (𝑥 ∈ On → (𝑥 ⊆ (ℵ‘𝑥) ↔ ¬ (ℵ‘𝑥) ∈ 𝑥))
21	18, 20	syl5ibr 248	. . 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 7574	1 ⊢ (𝐴 ∈ On → 𝐴 ⊆ (ℵ‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140   ⊆ wss 3938  Oncon0 6193  ‘cfv 6357  ℵcale 9367

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-inf2 9106

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-se 5517  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-isom 6366  df-riota 7116  df-om 7583  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-oi 8976  df-har 9024  df-card 9370  df-aleph 9371

This theorem is referenced by:  cardaleph  9517  alephfp  9536  winafp  10121