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Mirrors > Home > MPE Home > Th. List > alephmul | Structured version Visualization version GIF version |
Description: The product of two alephs is their maximum. Equation 6.1 of [Jech] p. 42. (Contributed by NM, 29-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
alephmul | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((ℵ‘𝐴) × (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alephgeom 9502 | . . . 4 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴)) | |
2 | fvex 6678 | . . . . 5 ⊢ (ℵ‘𝐴) ∈ V | |
3 | ssdomg 8549 | . . . . 5 ⊢ ((ℵ‘𝐴) ∈ V → (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴))) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴)) |
5 | 1, 4 | sylbi 219 | . . 3 ⊢ (𝐴 ∈ On → ω ≼ (ℵ‘𝐴)) |
6 | alephon 9489 | . . . 4 ⊢ (ℵ‘𝐴) ∈ On | |
7 | onenon 9372 | . . . 4 ⊢ ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card) | |
8 | 6, 7 | ax-mp 5 | . . 3 ⊢ (ℵ‘𝐴) ∈ dom card |
9 | 5, 8 | jctil 522 | . 2 ⊢ (𝐴 ∈ On → ((ℵ‘𝐴) ∈ dom card ∧ ω ≼ (ℵ‘𝐴))) |
10 | alephgeom 9502 | . . . 4 ⊢ (𝐵 ∈ On ↔ ω ⊆ (ℵ‘𝐵)) | |
11 | fvex 6678 | . . . . . 6 ⊢ (ℵ‘𝐵) ∈ V | |
12 | ssdomg 8549 | . . . . . 6 ⊢ ((ℵ‘𝐵) ∈ V → (ω ⊆ (ℵ‘𝐵) → ω ≼ (ℵ‘𝐵))) | |
13 | 11, 12 | ax-mp 5 | . . . . 5 ⊢ (ω ⊆ (ℵ‘𝐵) → ω ≼ (ℵ‘𝐵)) |
14 | infn0 8774 | . . . . 5 ⊢ (ω ≼ (ℵ‘𝐵) → (ℵ‘𝐵) ≠ ∅) | |
15 | 13, 14 | syl 17 | . . . 4 ⊢ (ω ⊆ (ℵ‘𝐵) → (ℵ‘𝐵) ≠ ∅) |
16 | 10, 15 | sylbi 219 | . . 3 ⊢ (𝐵 ∈ On → (ℵ‘𝐵) ≠ ∅) |
17 | alephon 9489 | . . . 4 ⊢ (ℵ‘𝐵) ∈ On | |
18 | onenon 9372 | . . . 4 ⊢ ((ℵ‘𝐵) ∈ On → (ℵ‘𝐵) ∈ dom card) | |
19 | 17, 18 | ax-mp 5 | . . 3 ⊢ (ℵ‘𝐵) ∈ dom card |
20 | 16, 19 | jctil 522 | . 2 ⊢ (𝐵 ∈ On → ((ℵ‘𝐵) ∈ dom card ∧ (ℵ‘𝐵) ≠ ∅)) |
21 | infxp 9631 | . 2 ⊢ ((((ℵ‘𝐴) ∈ dom card ∧ ω ≼ (ℵ‘𝐴)) ∧ ((ℵ‘𝐵) ∈ dom card ∧ (ℵ‘𝐵) ≠ ∅)) → ((ℵ‘𝐴) × (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵))) | |
22 | 9, 20, 21 | syl2an 597 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((ℵ‘𝐴) × (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 ≠ wne 3016 Vcvv 3495 ∪ cun 3934 ⊆ wss 3936 ∅c0 4291 class class class wbr 5059 × cxp 5548 dom cdm 5550 Oncon0 6186 ‘cfv 6350 ωcom 7574 ≈ cen 8500 ≼ cdom 8501 cardccrd 9358 ℵcale 9359 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-oi 8968 df-har 9016 df-dju 9324 df-card 9362 df-aleph 9363 |
This theorem is referenced by: (None) |
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