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Mirrors > Home > MPE Home > Th. List > alephfplem3 | Structured version Visualization version GIF version |
Description: Lemma for alephfp 9536. (Contributed by NM, 6-Nov-2004.) |
Ref | Expression |
---|---|
alephfplem.1 | ⊢ 𝐻 = (rec(ℵ, ω) ↾ ω) |
Ref | Expression |
---|---|
alephfplem3 | ⊢ (𝑣 ∈ ω → (𝐻‘𝑣) ∈ ran ℵ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6672 | . . 3 ⊢ (𝑣 = ∅ → (𝐻‘𝑣) = (𝐻‘∅)) | |
2 | 1 | eleq1d 2899 | . 2 ⊢ (𝑣 = ∅ → ((𝐻‘𝑣) ∈ ran ℵ ↔ (𝐻‘∅) ∈ ran ℵ)) |
3 | fveq2 6672 | . . 3 ⊢ (𝑣 = 𝑤 → (𝐻‘𝑣) = (𝐻‘𝑤)) | |
4 | 3 | eleq1d 2899 | . 2 ⊢ (𝑣 = 𝑤 → ((𝐻‘𝑣) ∈ ran ℵ ↔ (𝐻‘𝑤) ∈ ran ℵ)) |
5 | fveq2 6672 | . . 3 ⊢ (𝑣 = suc 𝑤 → (𝐻‘𝑣) = (𝐻‘suc 𝑤)) | |
6 | 5 | eleq1d 2899 | . 2 ⊢ (𝑣 = suc 𝑤 → ((𝐻‘𝑣) ∈ ran ℵ ↔ (𝐻‘suc 𝑤) ∈ ran ℵ)) |
7 | alephfplem.1 | . . 3 ⊢ 𝐻 = (rec(ℵ, ω) ↾ ω) | |
8 | 7 | alephfplem1 9532 | . 2 ⊢ (𝐻‘∅) ∈ ran ℵ |
9 | alephfnon 9493 | . . . 4 ⊢ ℵ Fn On | |
10 | alephsson 9528 | . . . . 5 ⊢ ran ℵ ⊆ On | |
11 | 10 | sseli 3965 | . . . 4 ⊢ ((𝐻‘𝑤) ∈ ran ℵ → (𝐻‘𝑤) ∈ On) |
12 | fnfvelrn 6850 | . . . 4 ⊢ ((ℵ Fn On ∧ (𝐻‘𝑤) ∈ On) → (ℵ‘(𝐻‘𝑤)) ∈ ran ℵ) | |
13 | 9, 11, 12 | sylancr 589 | . . 3 ⊢ ((𝐻‘𝑤) ∈ ran ℵ → (ℵ‘(𝐻‘𝑤)) ∈ ran ℵ) |
14 | 7 | alephfplem2 9533 | . . . 4 ⊢ (𝑤 ∈ ω → (𝐻‘suc 𝑤) = (ℵ‘(𝐻‘𝑤))) |
15 | 14 | eleq1d 2899 | . . 3 ⊢ (𝑤 ∈ ω → ((𝐻‘suc 𝑤) ∈ ran ℵ ↔ (ℵ‘(𝐻‘𝑤)) ∈ ran ℵ)) |
16 | 13, 15 | syl5ibr 248 | . 2 ⊢ (𝑤 ∈ ω → ((𝐻‘𝑤) ∈ ran ℵ → (𝐻‘suc 𝑤) ∈ ran ℵ)) |
17 | 2, 4, 6, 8, 16 | finds1 7613 | 1 ⊢ (𝑣 ∈ ω → (𝐻‘𝑣) ∈ ran ℵ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∅c0 4293 ran crn 5558 ↾ cres 5559 Oncon0 6193 suc csuc 6195 Fn wfn 6352 ‘cfv 6357 ωcom 7582 reccrdg 8047 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 ax-inf2 9106 |
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-rmo 3148 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-int 4879 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-se 5517 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-isom 6366 df-riota 7116 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-oi 8976 df-har 9024 df-card 9370 df-aleph 9371 |
This theorem is referenced by: alephfplem4 9535 alephfp 9536 |
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