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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 10140, 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 6902 | . . 3 ⊢ (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦)) | |
3 | 1, 2 | sseq12d 4015 | . 2 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ (ℵ‘𝑥) ↔ 𝑦 ⊆ (ℵ‘𝑦))) |
4 | id 22 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
5 | fveq2 6902 | . . 3 ⊢ (𝑥 = 𝐴 → (ℵ‘𝑥) = (ℵ‘𝐴)) | |
6 | 4, 5 | sseq12d 4015 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ (ℵ‘𝑥) ↔ 𝐴 ⊆ (ℵ‘𝐴))) |
7 | alephord2i 10108 | . . . . . 6 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → (ℵ‘𝑦) ∈ (ℵ‘𝑥))) | |
8 | 7 | imp 405 | . . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (ℵ‘𝑦) ∈ (ℵ‘𝑥)) |
9 | onelon 6399 | . . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On) | |
10 | alephon 10100 | . . . . . 6 ⊢ (ℵ‘𝑥) ∈ On | |
11 | ontr2 6421 | . . . . . 6 ⊢ ((𝑦 ∈ On ∧ (ℵ‘𝑥) ∈ On) → ((𝑦 ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝑥)) → 𝑦 ∈ (ℵ‘𝑥))) | |
12 | 9, 10, 11 | sylancl 584 | . . . . 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 3164 | . . 3 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝑦 ⊆ (ℵ‘𝑦) → ∀𝑦 ∈ 𝑥 𝑦 ∈ (ℵ‘𝑥))) |
15 | 10 | onirri 6487 | . . . . 5 ⊢ ¬ (ℵ‘𝑥) ∈ (ℵ‘𝑥) |
16 | eleq1 2817 | . . . . . 6 ⊢ (𝑦 = (ℵ‘𝑥) → (𝑦 ∈ (ℵ‘𝑥) ↔ (ℵ‘𝑥) ∈ (ℵ‘𝑥))) | |
17 | 16 | rspccv 3608 | . . . . 5 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ∈ (ℵ‘𝑥) → ((ℵ‘𝑥) ∈ 𝑥 → (ℵ‘𝑥) ∈ (ℵ‘𝑥))) |
18 | 15, 17 | mtoi 198 | . . . 4 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ∈ (ℵ‘𝑥) → ¬ (ℵ‘𝑥) ∈ 𝑥) |
19 | ontri1 6408 | . . . . 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 | imbitrrid 245 | . . 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 7868 | 1 ⊢ (𝐴 ∈ On → 𝐴 ⊆ (ℵ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3058 ⊆ wss 3949 Oncon0 6374 ‘cfv 6553 ℵcale 9967 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-oi 9541 df-har 9588 df-card 9970 df-aleph 9971 |
This theorem is referenced by: cardaleph 10120 alephfp 10139 winafp 10728 |
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