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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 4006 | . 2 ⊢ On ⊆ On | |
| 2 | ordon 7797 | . 2 ⊢ Ord On | |
| 3 | alephord2i 10117 | . . . 4 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → (ℵ‘𝑦) ∈ (ℵ‘𝑥))) | |
| 4 | 3 | ralrimiv 3145 | . . 3 ⊢ (𝑥 ∈ On → ∀𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ (ℵ‘𝑥)) |
| 5 | 4 | rgen 3063 | . 2 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 (ℵ‘𝑦) ∈ (ℵ‘𝑥) |
| 6 | alephfnon 10105 | . . . 4 ⊢ ℵ Fn On | |
| 7 | alephsson 10140 | . . . 4 ⊢ ran ℵ ⊆ On | |
| 8 | df-f 6565 | . . . 4 ⊢ (ℵ:On⟶On ↔ (ℵ Fn On ∧ ran ℵ ⊆ On)) | |
| 9 | 6, 7, 8 | mpbir2an 711 | . . 3 ⊢ ℵ:On⟶On |
| 10 | issmo2 8389 | . . 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 1463 | 1 ⊢ Smo ℵ |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 ran crn 5686 Ord word 6383 Oncon0 6384 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 Smo wsmo 8385 ℵcale 9976 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-smo 8386 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-oi 9550 df-har 9597 df-card 9979 df-aleph 9980 |
| This theorem is referenced by: alephf1ALT 10143 alephsing 10316 |
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