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Mirrors > Home > MPE Home > Th. List > alephle | Structured version Visualization version GIF version |
Description: The argument of the aleph function is less than or equal to its value. Exercise 2 of [TakeutiZaring] p. 91. (Later, in alephfp2 9538, we will that equality can sometimes hold.) (Contributed by NM, 9-Nov-2003.) (Proof shortened by Mario Carneiro, 22-Feb-2013.) |
Ref | Expression |
---|---|
alephle | ⊢ (𝐴 ∈ On → 𝐴 ⊆ (ℵ‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
2 | fveq2 6673 | . . 3 ⊢ (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦)) | |
3 | 1, 2 | sseq12d 4003 | . 2 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ (ℵ‘𝑥) ↔ 𝑦 ⊆ (ℵ‘𝑦))) |
4 | id 22 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
5 | fveq2 6673 | . . 3 ⊢ (𝑥 = 𝐴 → (ℵ‘𝑥) = (ℵ‘𝐴)) | |
6 | 4, 5 | sseq12d 4003 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ (ℵ‘𝑥) ↔ 𝐴 ⊆ (ℵ‘𝐴))) |
7 | alephord2i 9506 | . . . . . 6 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → (ℵ‘𝑦) ∈ (ℵ‘𝑥))) | |
8 | 7 | imp 409 | . . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (ℵ‘𝑦) ∈ (ℵ‘𝑥)) |
9 | onelon 6219 | . . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On) | |
10 | alephon 9498 | . . . . . 6 ⊢ (ℵ‘𝑥) ∈ On | |
11 | ontr2 6241 | . . . . . 6 ⊢ ((𝑦 ∈ On ∧ (ℵ‘𝑥) ∈ On) → ((𝑦 ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝑥)) → 𝑦 ∈ (ℵ‘𝑥))) | |
12 | 9, 10, 11 | sylancl 588 | . . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ((𝑦 ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝑥)) → 𝑦 ∈ (ℵ‘𝑥))) |
13 | 8, 12 | mpan2d 692 | . . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑦 ⊆ (ℵ‘𝑦) → 𝑦 ∈ (ℵ‘𝑥))) |
14 | 13 | ralimdva 3180 | . . 3 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝑦 ⊆ (ℵ‘𝑦) → ∀𝑦 ∈ 𝑥 𝑦 ∈ (ℵ‘𝑥))) |
15 | 10 | onirri 6300 | . . . . 5 ⊢ ¬ (ℵ‘𝑥) ∈ (ℵ‘𝑥) |
16 | eleq1 2903 | . . . . . 6 ⊢ (𝑦 = (ℵ‘𝑥) → (𝑦 ∈ (ℵ‘𝑥) ↔ (ℵ‘𝑥) ∈ (ℵ‘𝑥))) | |
17 | 16 | rspccv 3623 | . . . . 5 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ∈ (ℵ‘𝑥) → ((ℵ‘𝑥) ∈ 𝑥 → (ℵ‘𝑥) ∈ (ℵ‘𝑥))) |
18 | 15, 17 | mtoi 201 | . . . 4 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ∈ (ℵ‘𝑥) → ¬ (ℵ‘𝑥) ∈ 𝑥) |
19 | ontri1 6228 | . . . . 5 ⊢ ((𝑥 ∈ On ∧ (ℵ‘𝑥) ∈ On) → (𝑥 ⊆ (ℵ‘𝑥) ↔ ¬ (ℵ‘𝑥) ∈ 𝑥)) | |
20 | 10, 19 | mpan2 689 | . . . 4 ⊢ (𝑥 ∈ On → (𝑥 ⊆ (ℵ‘𝑥) ↔ ¬ (ℵ‘𝑥) ∈ 𝑥)) |
21 | 18, 20 | syl5ibr 248 | . . 3 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝑦 ∈ (ℵ‘𝑥) → 𝑥 ⊆ (ℵ‘𝑥))) |
22 | 14, 21 | syld 47 | . 2 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝑦 ⊆ (ℵ‘𝑦) → 𝑥 ⊆ (ℵ‘𝑥))) |
23 | 3, 6, 22 | tfis3 7575 | 1 ⊢ (𝐴 ∈ On → 𝐴 ⊆ (ℵ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3141 ⊆ wss 3939 Oncon0 6194 ‘cfv 6358 ℵcale 9368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-oi 8977 df-har 9025 df-card 9371 df-aleph 9372 |
This theorem is referenced by: cardaleph 9518 alephfp 9537 winafp 10122 |
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