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Theorem alephle 8767
Description: The argument of the aleph function is less than or equal to its value. Exercise 2 of [TakeutiZaring] p. 91. (Later, in alephfp2 8788, 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 6084 . . 3 (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
31, 2sseq12d 3592 . 2 (𝑥 = 𝑦 → (𝑥 ⊆ (ℵ‘𝑥) ↔ 𝑦 ⊆ (ℵ‘𝑦)))
4 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
5 fveq2 6084 . . 3 (𝑥 = 𝐴 → (ℵ‘𝑥) = (ℵ‘𝐴))
64, 5sseq12d 3592 . 2 (𝑥 = 𝐴 → (𝑥 ⊆ (ℵ‘𝑥) ↔ 𝐴 ⊆ (ℵ‘𝐴)))
7 alephord2i 8756 . . . . . 6 (𝑥 ∈ On → (𝑦𝑥 → (ℵ‘𝑦) ∈ (ℵ‘𝑥)))
87imp 443 . . . . 5 ((𝑥 ∈ On ∧ 𝑦𝑥) → (ℵ‘𝑦) ∈ (ℵ‘𝑥))
9 onelon 5647 . . . . . 6 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
10 alephon 8748 . . . . . 6 (ℵ‘𝑥) ∈ On
11 ontr2 5671 . . . . . 6 ((𝑦 ∈ On ∧ (ℵ‘𝑥) ∈ On) → ((𝑦 ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝑥)) → 𝑦 ∈ (ℵ‘𝑥)))
129, 10, 11sylancl 692 . . . . 5 ((𝑥 ∈ On ∧ 𝑦𝑥) → ((𝑦 ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝑥)) → 𝑦 ∈ (ℵ‘𝑥)))
138, 12mpan2d 705 . . . 4 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑦 ⊆ (ℵ‘𝑦) → 𝑦 ∈ (ℵ‘𝑥)))
1413ralimdva 2940 . . 3 (𝑥 ∈ On → (∀𝑦𝑥 𝑦 ⊆ (ℵ‘𝑦) → ∀𝑦𝑥 𝑦 ∈ (ℵ‘𝑥)))
1510onirri 5733 . . . . 5 ¬ (ℵ‘𝑥) ∈ (ℵ‘𝑥)
16 eleq1 2671 . . . . . 6 (𝑦 = (ℵ‘𝑥) → (𝑦 ∈ (ℵ‘𝑥) ↔ (ℵ‘𝑥) ∈ (ℵ‘𝑥)))
1716rspccv 3274 . . . . 5 (∀𝑦𝑥 𝑦 ∈ (ℵ‘𝑥) → ((ℵ‘𝑥) ∈ 𝑥 → (ℵ‘𝑥) ∈ (ℵ‘𝑥)))
1815, 17mtoi 188 . . . 4 (∀𝑦𝑥 𝑦 ∈ (ℵ‘𝑥) → ¬ (ℵ‘𝑥) ∈ 𝑥)
19 ontri1 5656 . . . . 5 ((𝑥 ∈ On ∧ (ℵ‘𝑥) ∈ On) → (𝑥 ⊆ (ℵ‘𝑥) ↔ ¬ (ℵ‘𝑥) ∈ 𝑥))
2010, 19mpan2 702 . . . 4 (𝑥 ∈ On → (𝑥 ⊆ (ℵ‘𝑥) ↔ ¬ (ℵ‘𝑥) ∈ 𝑥))
2118, 20syl5ibr 234 . . 3 (𝑥 ∈ On → (∀𝑦𝑥 𝑦 ∈ (ℵ‘𝑥) → 𝑥 ⊆ (ℵ‘𝑥)))
2214, 21syld 45 . 2 (𝑥 ∈ On → (∀𝑦𝑥 𝑦 ⊆ (ℵ‘𝑦) → 𝑥 ⊆ (ℵ‘𝑥)))
233, 6, 22tfis3 6922 1 (𝐴 ∈ On → 𝐴 ⊆ (ℵ‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wcel 1975  wral 2891  wss 3535  Oncon0 5622  cfv 5786  cale 8618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-inf2 8394
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-se 4984  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-isom 5795  df-riota 6485  df-om 6931  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-er 7602  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-oi 8271  df-har 8319  df-card 8621  df-aleph 8622
This theorem is referenced by:  cardaleph  8768  alephfp  8787  winafp  9371
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