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Theorem alephle 10082
Description: The argument of the aleph function is less than or equal to its value. Exercise 2 of [TakeutiZaring] p. 91. (Later, in alephfp2 10103, 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 6884 . . 3 (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
31, 2sseq12d 4010 . 2 (𝑥 = 𝑦 → (𝑥 ⊆ (ℵ‘𝑥) ↔ 𝑦 ⊆ (ℵ‘𝑦)))
4 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
5 fveq2 6884 . . 3 (𝑥 = 𝐴 → (ℵ‘𝑥) = (ℵ‘𝐴))
64, 5sseq12d 4010 . 2 (𝑥 = 𝐴 → (𝑥 ⊆ (ℵ‘𝑥) ↔ 𝐴 ⊆ (ℵ‘𝐴)))
7 alephord2i 10071 . . . . . 6 (𝑥 ∈ On → (𝑦𝑥 → (ℵ‘𝑦) ∈ (ℵ‘𝑥)))
87imp 406 . . . . 5 ((𝑥 ∈ On ∧ 𝑦𝑥) → (ℵ‘𝑦) ∈ (ℵ‘𝑥))
9 onelon 6382 . . . . . 6 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
10 alephon 10063 . . . . . 6 (ℵ‘𝑥) ∈ On
11 ontr2 6404 . . . . . 6 ((𝑦 ∈ On ∧ (ℵ‘𝑥) ∈ On) → ((𝑦 ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝑥)) → 𝑦 ∈ (ℵ‘𝑥)))
129, 10, 11sylancl 585 . . . . 5 ((𝑥 ∈ On ∧ 𝑦𝑥) → ((𝑦 ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝑥)) → 𝑦 ∈ (ℵ‘𝑥)))
138, 12mpan2d 691 . . . 4 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑦 ⊆ (ℵ‘𝑦) → 𝑦 ∈ (ℵ‘𝑥)))
1413ralimdva 3161 . . 3 (𝑥 ∈ On → (∀𝑦𝑥 𝑦 ⊆ (ℵ‘𝑦) → ∀𝑦𝑥 𝑦 ∈ (ℵ‘𝑥)))
1510onirri 6470 . . . . 5 ¬ (ℵ‘𝑥) ∈ (ℵ‘𝑥)
16 eleq1 2815 . . . . . 6 (𝑦 = (ℵ‘𝑥) → (𝑦 ∈ (ℵ‘𝑥) ↔ (ℵ‘𝑥) ∈ (ℵ‘𝑥)))
1716rspccv 3603 . . . . 5 (∀𝑦𝑥 𝑦 ∈ (ℵ‘𝑥) → ((ℵ‘𝑥) ∈ 𝑥 → (ℵ‘𝑥) ∈ (ℵ‘𝑥)))
1815, 17mtoi 198 . . . 4 (∀𝑦𝑥 𝑦 ∈ (ℵ‘𝑥) → ¬ (ℵ‘𝑥) ∈ 𝑥)
19 ontri1 6391 . . . . 5 ((𝑥 ∈ On ∧ (ℵ‘𝑥) ∈ On) → (𝑥 ⊆ (ℵ‘𝑥) ↔ ¬ (ℵ‘𝑥) ∈ 𝑥))
2010, 19mpan2 688 . . . 4 (𝑥 ∈ On → (𝑥 ⊆ (ℵ‘𝑥) ↔ ¬ (ℵ‘𝑥) ∈ 𝑥))
2118, 20imbitrrid 245 . . 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 7843 1 (𝐴 ∈ On → 𝐴 ⊆ (ℵ‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wral 3055  wss 3943  Oncon0 6357  cfv 6536  cale 9930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-inf2 9635
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7360  df-ov 7407  df-om 7852  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-1o 8464  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-oi 9504  df-har 9551  df-card 9933  df-aleph 9934
This theorem is referenced by:  cardaleph  10083  alephfp  10102  winafp  10691
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