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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 10120 | . . . 4 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴)) | |
2 | fvex 6920 | . . . . 5 ⊢ (ℵ‘𝐴) ∈ V | |
3 | ssdomg 9039 | . . . . 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 10107 | . . . 4 ⊢ (ℵ‘𝐴) ∈ On | |
7 | onenon 9987 | . . . 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 10120 | . . . 4 ⊢ (𝐵 ∈ On ↔ ω ⊆ (ℵ‘𝐵)) | |
11 | fvex 6920 | . . . . . 6 ⊢ (ℵ‘𝐵) ∈ V | |
12 | ssdomg 9039 | . . . . . 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 9338 | . . . . 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 10107 | . . . 4 ⊢ (ℵ‘𝐵) ∈ On | |
18 | onenon 9987 | . . . 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 10252 | . 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 2106 ≠ wne 2938 Vcvv 3478 ∪ cun 3961 ⊆ wss 3963 ∅c0 4339 class class class wbr 5148 × cxp 5687 dom cdm 5689 Oncon0 6386 ‘cfv 6563 ωcom 7887 ≈ cen 8981 ≼ cdom 8982 cardccrd 9973 ℵcale 9974 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-oi 9548 df-har 9595 df-dju 9939 df-card 9977 df-aleph 9978 |
This theorem is referenced by: (None) |
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