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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 10122 | . . . 4 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴)) | |
| 2 | fvex 6919 | . . . . 5 ⊢ (ℵ‘𝐴) ∈ V | |
| 3 | ssdomg 9040 | . . . . 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 217 | . . 3 ⊢ (𝐴 ∈ On → ω ≼ (ℵ‘𝐴)) |
| 6 | alephon 10109 | . . . 4 ⊢ (ℵ‘𝐴) ∈ On | |
| 7 | onenon 9989 | . . . 4 ⊢ ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card) | |
| 8 | 6, 7 | ax-mp 5 | . . 3 ⊢ (ℵ‘𝐴) ∈ dom card |
| 9 | 5, 8 | jctil 519 | . 2 ⊢ (𝐴 ∈ On → ((ℵ‘𝐴) ∈ dom card ∧ ω ≼ (ℵ‘𝐴))) |
| 10 | alephgeom 10122 | . . . 4 ⊢ (𝐵 ∈ On ↔ ω ⊆ (ℵ‘𝐵)) | |
| 11 | fvex 6919 | . . . . . 6 ⊢ (ℵ‘𝐵) ∈ V | |
| 12 | ssdomg 9040 | . . . . . 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 9340 | . . . . 5 ⊢ (ω ≼ (ℵ‘𝐵) → (ℵ‘𝐵) ≠ ∅) | |
| 15 | 13, 14 | syl 17 | . . . 4 ⊢ (ω ⊆ (ℵ‘𝐵) → (ℵ‘𝐵) ≠ ∅) |
| 16 | 10, 15 | sylbi 217 | . . 3 ⊢ (𝐵 ∈ On → (ℵ‘𝐵) ≠ ∅) |
| 17 | alephon 10109 | . . . 4 ⊢ (ℵ‘𝐵) ∈ On | |
| 18 | onenon 9989 | . . . 4 ⊢ ((ℵ‘𝐵) ∈ On → (ℵ‘𝐵) ∈ dom card) | |
| 19 | 17, 18 | ax-mp 5 | . . 3 ⊢ (ℵ‘𝐵) ∈ dom card |
| 20 | 16, 19 | jctil 519 | . 2 ⊢ (𝐵 ∈ On → ((ℵ‘𝐵) ∈ dom card ∧ (ℵ‘𝐵) ≠ ∅)) |
| 21 | infxp 10254 | . 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 596 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((ℵ‘𝐴) × (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∪ cun 3949 ⊆ wss 3951 ∅c0 4333 class class class wbr 5143 × cxp 5683 dom cdm 5685 Oncon0 6384 ‘cfv 6561 ωcom 7887 ≈ cen 8982 ≼ cdom 8983 cardccrd 9975 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-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-oi 9550 df-har 9597 df-dju 9941 df-card 9979 df-aleph 9980 |
| This theorem is referenced by: (None) |
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