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Theorem alephle 9958
Description: The argument of the aleph function is less than or equal to its value. Exercise 2 of [TakeutiZaring] p. 91. (Later, in alephfp2 9979, 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 6838 . . 3 (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
31, 2sseq12d 3976 . 2 (𝑥 = 𝑦 → (𝑥 ⊆ (ℵ‘𝑥) ↔ 𝑦 ⊆ (ℵ‘𝑦)))
4 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
5 fveq2 6838 . . 3 (𝑥 = 𝐴 → (ℵ‘𝑥) = (ℵ‘𝐴))
64, 5sseq12d 3976 . 2 (𝑥 = 𝐴 → (𝑥 ⊆ (ℵ‘𝑥) ↔ 𝐴 ⊆ (ℵ‘𝐴)))
7 alephord2i 9947 . . . . . 6 (𝑥 ∈ On → (𝑦𝑥 → (ℵ‘𝑦) ∈ (ℵ‘𝑥)))
87imp 408 . . . . 5 ((𝑥 ∈ On ∧ 𝑦𝑥) → (ℵ‘𝑦) ∈ (ℵ‘𝑥))
9 onelon 6339 . . . . . 6 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
10 alephon 9939 . . . . . 6 (ℵ‘𝑥) ∈ On
11 ontr2 6361 . . . . . 6 ((𝑦 ∈ On ∧ (ℵ‘𝑥) ∈ On) → ((𝑦 ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝑥)) → 𝑦 ∈ (ℵ‘𝑥)))
129, 10, 11sylancl 587 . . . . 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 3163 . . 3 (𝑥 ∈ On → (∀𝑦𝑥 𝑦 ⊆ (ℵ‘𝑦) → ∀𝑦𝑥 𝑦 ∈ (ℵ‘𝑥)))
1510onirri 6426 . . . . 5 ¬ (ℵ‘𝑥) ∈ (ℵ‘𝑥)
16 eleq1 2826 . . . . . 6 (𝑦 = (ℵ‘𝑥) → (𝑦 ∈ (ℵ‘𝑥) ↔ (ℵ‘𝑥) ∈ (ℵ‘𝑥)))
1716rspccv 3577 . . . . 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 6348 . . . . 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 246 . . 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 7785 1 (𝐴 ∈ On → 𝐴 ⊆ (ℵ‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3063  wss 3909  Oncon0 6314  cfv 6492  cale 9806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7663  ax-inf2 9511
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-int 4907  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-tr 5222  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-se 5587  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6250  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7306  df-ov 7353  df-om 7794  df-2nd 7913  df-frecs 8180  df-wrecs 8211  df-recs 8285  df-rdg 8324  df-1o 8380  df-er 8582  df-en 8818  df-dom 8819  df-sdom 8820  df-fin 8821  df-oi 9380  df-har 9427  df-card 9809  df-aleph 9810
This theorem is referenced by:  cardaleph  9959  alephfp  9978  winafp  10567
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