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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 4001 | . 2 ⊢ On ⊆ On | |
2 | ordon 7774 | . 2 ⊢ Ord On | |
3 | alephord2i 10110 | . . . 4 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → (ℵ‘𝑦) ∈ (ℵ‘𝑥))) | |
4 | 3 | ralrimiv 3135 | . . 3 ⊢ (𝑥 ∈ On → ∀𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ (ℵ‘𝑥)) |
5 | 4 | rgen 3053 | . 2 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ (ℵ‘𝑥) |
6 | alephfnon 10098 | . . . 4 ⊢ ℵ Fn On | |
7 | alephsson 10133 | . . . 4 ⊢ ran ℵ ⊆ On | |
8 | df-f 6547 | . . . 4 ⊢ (ℵ:On⟶On ↔ (ℵ Fn On ∧ ran ℵ ⊆ On)) | |
9 | 6, 7, 8 | mpbir2an 709 | . . 3 ⊢ ℵ:On⟶On |
10 | issmo2 8368 | . . 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 1458 | 1 ⊢ Smo ℵ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 ∈ wcel 2099 ∀wral 3051 ⊆ wss 3946 ran crn 5673 Ord word 6364 Oncon0 6365 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 Smo wsmo 8364 ℵcale 9969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-inf2 9674 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6302 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-om 7866 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-smo 8365 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-oi 9543 df-har 9590 df-card 9972 df-aleph 9973 |
This theorem is referenced by: alephf1ALT 10136 alephsing 10307 |
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