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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 3880 | . 2 ⊢ On ⊆ On | |
2 | ordon 7314 | . 2 ⊢ Ord On | |
3 | alephord2i 9297 | . . . 4 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → (ℵ‘𝑦) ∈ (ℵ‘𝑥))) | |
4 | 3 | ralrimiv 3132 | . . 3 ⊢ (𝑥 ∈ On → ∀𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ (ℵ‘𝑥)) |
5 | 4 | rgen 3099 | . 2 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ (ℵ‘𝑥) |
6 | alephfnon 9285 | . . . 4 ⊢ ℵ Fn On | |
7 | alephsson 9320 | . . . 4 ⊢ ran ℵ ⊆ On | |
8 | df-f 6192 | . . . 4 ⊢ (ℵ:On⟶On ↔ (ℵ Fn On ∧ ran ℵ ⊆ On)) | |
9 | 6, 7, 8 | mpbir2an 698 | . . 3 ⊢ ℵ:On⟶On |
10 | issmo2 7790 | . . 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 1440 | 1 ⊢ Smo ℵ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1068 ∈ wcel 2050 ∀wral 3089 ⊆ wss 3830 ran crn 5408 Ord word 6028 Oncon0 6029 Fn wfn 6183 ⟶wf 6184 ‘cfv 6188 Smo wsmo 7786 ℵcale 9159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-inf2 8898 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-se 5367 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-isom 6197 df-riota 6937 df-om 7397 df-wrecs 7750 df-smo 7787 df-recs 7812 df-rdg 7850 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-oi 8769 df-har 8817 df-card 9162 df-aleph 9163 |
This theorem is referenced by: alephf1ALT 9323 alephsing 9496 |
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