Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 8932, 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 6191 | . . 3 ⊢ (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦)) | |
3 | 1, 2 | sseq12d 3634 | . 2 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ (ℵ‘𝑥) ↔ 𝑦 ⊆ (ℵ‘𝑦))) |
4 | id 22 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
5 | fveq2 6191 | . . 3 ⊢ (𝑥 = 𝐴 → (ℵ‘𝑥) = (ℵ‘𝐴)) | |
6 | 4, 5 | sseq12d 3634 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ (ℵ‘𝑥) ↔ 𝐴 ⊆ (ℵ‘𝐴))) |
7 | alephord2i 8900 | . . . . . 6 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → (ℵ‘𝑦) ∈ (ℵ‘𝑥))) | |
8 | 7 | imp 445 | . . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (ℵ‘𝑦) ∈ (ℵ‘𝑥)) |
9 | onelon 5748 | . . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On) | |
10 | alephon 8892 | . . . . . 6 ⊢ (ℵ‘𝑥) ∈ On | |
11 | ontr2 5772 | . . . . . 6 ⊢ ((𝑦 ∈ On ∧ (ℵ‘𝑥) ∈ On) → ((𝑦 ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝑥)) → 𝑦 ∈ (ℵ‘𝑥))) | |
12 | 9, 10, 11 | sylancl 694 | . . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ((𝑦 ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝑥)) → 𝑦 ∈ (ℵ‘𝑥))) |
13 | 8, 12 | mpan2d 710 | . . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (𝑦 ⊆ (ℵ‘𝑦) → 𝑦 ∈ (ℵ‘𝑥))) |
14 | 13 | ralimdva 2962 | . . 3 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝑦 ⊆ (ℵ‘𝑦) → ∀𝑦 ∈ 𝑥 𝑦 ∈ (ℵ‘𝑥))) |
15 | 10 | onirri 5834 | . . . . 5 ⊢ ¬ (ℵ‘𝑥) ∈ (ℵ‘𝑥) |
16 | eleq1 2689 | . . . . . 6 ⊢ (𝑦 = (ℵ‘𝑥) → (𝑦 ∈ (ℵ‘𝑥) ↔ (ℵ‘𝑥) ∈ (ℵ‘𝑥))) | |
17 | 16 | rspccv 3306 | . . . . 5 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ∈ (ℵ‘𝑥) → ((ℵ‘𝑥) ∈ 𝑥 → (ℵ‘𝑥) ∈ (ℵ‘𝑥))) |
18 | 15, 17 | mtoi 190 | . . . 4 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ∈ (ℵ‘𝑥) → ¬ (ℵ‘𝑥) ∈ 𝑥) |
19 | ontri1 5757 | . . . . 5 ⊢ ((𝑥 ∈ On ∧ (ℵ‘𝑥) ∈ On) → (𝑥 ⊆ (ℵ‘𝑥) ↔ ¬ (ℵ‘𝑥) ∈ 𝑥)) | |
20 | 10, 19 | mpan2 707 | . . . 4 ⊢ (𝑥 ∈ On → (𝑥 ⊆ (ℵ‘𝑥) ↔ ¬ (ℵ‘𝑥) ∈ 𝑥)) |
21 | 18, 20 | syl5ibr 236 | . . 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 7057 | 1 ⊢ (𝐴 ∈ On → 𝐴 ⊆ (ℵ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ⊆ wss 3574 Oncon0 5723 ‘cfv 5888 ℵcale 8762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-oi 8415 df-har 8463 df-card 8765 df-aleph 8766 |
This theorem is referenced by: cardaleph 8912 alephfp 8931 winafp 9519 |
Copyright terms: Public domain | W3C validator |