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Theorem alephle 9502
 Description: The argument of the aleph function is less than or equal to its value. Exercise 2 of [TakeutiZaring] p. 91. (Later, in alephfp2 9523, 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.
StepHypRef Expression
1 id 22 . . 3 (𝑥 = 𝑦𝑥 = 𝑦)
2 fveq2 6646 . . 3 (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
31, 2sseq12d 3948 . 2 (𝑥 = 𝑦 → (𝑥 ⊆ (ℵ‘𝑥) ↔ 𝑦 ⊆ (ℵ‘𝑦)))
4 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
5 fveq2 6646 . . 3 (𝑥 = 𝐴 → (ℵ‘𝑥) = (ℵ‘𝐴))
64, 5sseq12d 3948 . 2 (𝑥 = 𝐴 → (𝑥 ⊆ (ℵ‘𝑥) ↔ 𝐴 ⊆ (ℵ‘𝐴)))
7 alephord2i 9491 . . . . . 6 (𝑥 ∈ On → (𝑦𝑥 → (ℵ‘𝑦) ∈ (ℵ‘𝑥)))
87imp 410 . . . . 5 ((𝑥 ∈ On ∧ 𝑦𝑥) → (ℵ‘𝑦) ∈ (ℵ‘𝑥))
9 onelon 6185 . . . . . 6 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
10 alephon 9483 . . . . . 6 (ℵ‘𝑥) ∈ On
11 ontr2 6207 . . . . . 6 ((𝑦 ∈ On ∧ (ℵ‘𝑥) ∈ On) → ((𝑦 ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝑥)) → 𝑦 ∈ (ℵ‘𝑥)))
129, 10, 11sylancl 589 . . . . 5 ((𝑥 ∈ On ∧ 𝑦𝑥) → ((𝑦 ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝑥)) → 𝑦 ∈ (ℵ‘𝑥)))
138, 12mpan2d 693 . . . 4 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑦 ⊆ (ℵ‘𝑦) → 𝑦 ∈ (ℵ‘𝑥)))
1413ralimdva 3144 . . 3 (𝑥 ∈ On → (∀𝑦𝑥 𝑦 ⊆ (ℵ‘𝑦) → ∀𝑦𝑥 𝑦 ∈ (ℵ‘𝑥)))
1510onirri 6266 . . . . 5 ¬ (ℵ‘𝑥) ∈ (ℵ‘𝑥)
16 eleq1 2877 . . . . . 6 (𝑦 = (ℵ‘𝑥) → (𝑦 ∈ (ℵ‘𝑥) ↔ (ℵ‘𝑥) ∈ (ℵ‘𝑥)))
1716rspccv 3568 . . . . 5 (∀𝑦𝑥 𝑦 ∈ (ℵ‘𝑥) → ((ℵ‘𝑥) ∈ 𝑥 → (ℵ‘𝑥) ∈ (ℵ‘𝑥)))
1815, 17mtoi 202 . . . 4 (∀𝑦𝑥 𝑦 ∈ (ℵ‘𝑥) → ¬ (ℵ‘𝑥) ∈ 𝑥)
19 ontri1 6194 . . . . 5 ((𝑥 ∈ On ∧ (ℵ‘𝑥) ∈ On) → (𝑥 ⊆ (ℵ‘𝑥) ↔ ¬ (ℵ‘𝑥) ∈ 𝑥))
2010, 19mpan2 690 . . . 4 (𝑥 ∈ On → (𝑥 ⊆ (ℵ‘𝑥) ↔ ¬ (ℵ‘𝑥) ∈ 𝑥))
2118, 20syl5ibr 249 . . 3 (𝑥 ∈ On → (∀𝑦𝑥 𝑦 ∈ (ℵ‘𝑥) → 𝑥 ⊆ (ℵ‘𝑥)))
2214, 21syld 47 . 2 (𝑥 ∈ On → (∀𝑦𝑥 𝑦 ⊆ (ℵ‘𝑦) → 𝑥 ⊆ (ℵ‘𝑥)))
233, 6, 22tfis3 7555 1 (𝐴 ∈ On → 𝐴 ⊆ (ℵ‘𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ⊆ wss 3881  Oncon0 6160  ‘cfv 6325  ℵcale 9352 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-inf2 9091 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-se 5480  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-isom 6334  df-riota 7094  df-om 7564  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-er 8275  df-en 8496  df-dom 8497  df-sdom 8498  df-fin 8499  df-oi 8961  df-har 9008  df-card 9355  df-aleph 9356 This theorem is referenced by:  cardaleph  9503  alephfp  9522  winafp  10111
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