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Mirrors > Home > MPE Home > Th. List > alephlim | Structured version Visualization version GIF version |
Description: Value of the aleph function at a limit ordinal. Definition 12(iii) of [Suppes] p. 91. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 13-Sep-2013.) |
Ref | Expression |
---|---|
alephlim | ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (ℵ‘𝐴) = ∪ 𝑥 ∈ 𝐴 (ℵ‘𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rdglim2a 8071 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (rec(har, ω)‘𝐴) = ∪ 𝑥 ∈ 𝐴 (rec(har, ω)‘𝑥)) | |
2 | df-aleph 9371 | . . 3 ⊢ ℵ = rec(har, ω) | |
3 | 2 | fveq1i 6673 | . 2 ⊢ (ℵ‘𝐴) = (rec(har, ω)‘𝐴) |
4 | 2 | fveq1i 6673 | . . . 4 ⊢ (ℵ‘𝑥) = (rec(har, ω)‘𝑥) |
5 | 4 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (ℵ‘𝑥) = (rec(har, ω)‘𝑥)) |
6 | 5 | iuneq2i 4942 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 (ℵ‘𝑥) = ∪ 𝑥 ∈ 𝐴 (rec(har, ω)‘𝑥) |
7 | 1, 3, 6 | 3eqtr4g 2883 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (ℵ‘𝐴) = ∪ 𝑥 ∈ 𝐴 (ℵ‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∪ ciun 4921 Lim wlim 6194 ‘cfv 6357 ωcom 7582 reccrdg 8047 harchar 9022 ℵcale 9367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-aleph 9371 |
This theorem is referenced by: alephon 9497 alephcard 9498 alephordi 9502 cardaleph 9517 alephsing 9700 pwcfsdom 10007 |
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