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Mirrors > Home > MPE Home > Th. List > alephsmo | Structured version Visualization version GIF version |
Description: The aleph function is strictly monotone. (Contributed by Mario Carneiro, 15-Mar-2013.) |
Ref | Expression |
---|---|
alephsmo | ⊢ Smo ℵ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3986 | . 2 ⊢ On ⊆ On | |
2 | ordon 7487 | . 2 ⊢ Ord On | |
3 | alephord2i 9491 | . . . 4 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → (ℵ‘𝑦) ∈ (ℵ‘𝑥))) | |
4 | 3 | ralrimiv 3178 | . . 3 ⊢ (𝑥 ∈ On → ∀𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ (ℵ‘𝑥)) |
5 | 4 | rgen 3145 | . 2 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ (ℵ‘𝑥) |
6 | alephfnon 9479 | . . . 4 ⊢ ℵ Fn On | |
7 | alephsson 9514 | . . . 4 ⊢ ran ℵ ⊆ On | |
8 | df-f 6352 | . . . 4 ⊢ (ℵ:On⟶On ↔ (ℵ Fn On ∧ ran ℵ ⊆ On)) | |
9 | 6, 7, 8 | mpbir2an 707 | . . 3 ⊢ ℵ:On⟶On |
10 | issmo2 7975 | . . 3 ⊢ (ℵ:On⟶On → ((On ⊆ On ∧ Ord On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ (ℵ‘𝑥)) → Smo ℵ)) | |
11 | 9, 10 | ax-mp 5 | . 2 ⊢ ((On ⊆ On ∧ Ord On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ (ℵ‘𝑥)) → Smo ℵ) |
12 | 1, 2, 5, 11 | mp3an 1452 | 1 ⊢ Smo ℵ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 ∈ wcel 2105 ∀wral 3135 ⊆ wss 3933 ran crn 5549 Ord word 6183 Oncon0 6184 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 Smo wsmo 7971 ℵcale 9353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-om 7570 df-wrecs 7936 df-smo 7972 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-oi 8962 df-har 9010 df-card 9356 df-aleph 9357 |
This theorem is referenced by: alephf1ALT 9517 alephsing 9686 |
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