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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 9501 | . . . 4 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴)) | |
2 | fvex 6676 | . . . . 5 ⊢ (ℵ‘𝐴) ∈ V | |
3 | ssdomg 8548 | . . . . 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 9488 | . . . 4 ⊢ (ℵ‘𝐴) ∈ On | |
7 | onenon 9371 | . . . 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 9501 | . . . 4 ⊢ (𝐵 ∈ On ↔ ω ⊆ (ℵ‘𝐵)) | |
11 | fvex 6676 | . . . . . 6 ⊢ (ℵ‘𝐵) ∈ V | |
12 | ssdomg 8548 | . . . . . 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 8773 | . . . . 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 9488 | . . . 4 ⊢ (ℵ‘𝐵) ∈ On | |
18 | onenon 9371 | . . . 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 9630 | . 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 2113 ≠ wne 3015 Vcvv 3491 ∪ cun 3927 ⊆ wss 3929 ∅c0 4284 class class class wbr 5059 × cxp 5546 dom cdm 5548 Oncon0 6184 ‘cfv 6348 ωcom 7573 ≈ cen 8499 ≼ cdom 8500 cardccrd 9357 ℵcale 9358 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-inf2 9097 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 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 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-2o 8096 df-oadd 8099 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-oi 8967 df-har 9015 df-dju 9323 df-card 9361 df-aleph 9362 |
This theorem is referenced by: (None) |
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