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Theorem alephle 9360
Description: The argument of the aleph function is less than or equal to its value. Exercise 2 of [TakeutiZaring] p. 91. (Later, in alephfp2 9381, 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 6538 . . 3 (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
31, 2sseq12d 3921 . 2 (𝑥 = 𝑦 → (𝑥 ⊆ (ℵ‘𝑥) ↔ 𝑦 ⊆ (ℵ‘𝑦)))
4 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
5 fveq2 6538 . . 3 (𝑥 = 𝐴 → (ℵ‘𝑥) = (ℵ‘𝐴))
64, 5sseq12d 3921 . 2 (𝑥 = 𝐴 → (𝑥 ⊆ (ℵ‘𝑥) ↔ 𝐴 ⊆ (ℵ‘𝐴)))
7 alephord2i 9349 . . . . . 6 (𝑥 ∈ On → (𝑦𝑥 → (ℵ‘𝑦) ∈ (ℵ‘𝑥)))
87imp 407 . . . . 5 ((𝑥 ∈ On ∧ 𝑦𝑥) → (ℵ‘𝑦) ∈ (ℵ‘𝑥))
9 onelon 6091 . . . . . 6 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
10 alephon 9341 . . . . . 6 (ℵ‘𝑥) ∈ On
11 ontr2 6113 . . . . . 6 ((𝑦 ∈ On ∧ (ℵ‘𝑥) ∈ On) → ((𝑦 ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝑥)) → 𝑦 ∈ (ℵ‘𝑥)))
129, 10, 11sylancl 586 . . . . 5 ((𝑥 ∈ On ∧ 𝑦𝑥) → ((𝑦 ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝑥)) → 𝑦 ∈ (ℵ‘𝑥)))
138, 12mpan2d 690 . . . 4 ((𝑥 ∈ On ∧ 𝑦𝑥) → (𝑦 ⊆ (ℵ‘𝑦) → 𝑦 ∈ (ℵ‘𝑥)))
1413ralimdva 3144 . . 3 (𝑥 ∈ On → (∀𝑦𝑥 𝑦 ⊆ (ℵ‘𝑦) → ∀𝑦𝑥 𝑦 ∈ (ℵ‘𝑥)))
1510onirri 6172 . . . . 5 ¬ (ℵ‘𝑥) ∈ (ℵ‘𝑥)
16 eleq1 2870 . . . . . 6 (𝑦 = (ℵ‘𝑥) → (𝑦 ∈ (ℵ‘𝑥) ↔ (ℵ‘𝑥) ∈ (ℵ‘𝑥)))
1716rspccv 3556 . . . . 5 (∀𝑦𝑥 𝑦 ∈ (ℵ‘𝑥) → ((ℵ‘𝑥) ∈ 𝑥 → (ℵ‘𝑥) ∈ (ℵ‘𝑥)))
1815, 17mtoi 200 . . . 4 (∀𝑦𝑥 𝑦 ∈ (ℵ‘𝑥) → ¬ (ℵ‘𝑥) ∈ 𝑥)
19 ontri1 6100 . . . . 5 ((𝑥 ∈ On ∧ (ℵ‘𝑥) ∈ On) → (𝑥 ⊆ (ℵ‘𝑥) ↔ ¬ (ℵ‘𝑥) ∈ 𝑥))
2010, 19mpan2 687 . . . 4 (𝑥 ∈ On → (𝑥 ⊆ (ℵ‘𝑥) ↔ ¬ (ℵ‘𝑥) ∈ 𝑥))
2118, 20syl5ibr 247 . . 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 7428 1 (𝐴 ∈ On → 𝐴 ⊆ (ℵ‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1522  wcel 2081  wral 3105  wss 3859  Oncon0 6066  cfv 6225  cale 9211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-inf2 8950
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-om 7437  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-er 8139  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-oi 8820  df-har 8868  df-card 9214  df-aleph 9215
This theorem is referenced by:  cardaleph  9361  alephfp  9380  winafp  9965
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