![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > alephfplem3 | Structured version Visualization version GIF version |
Description: Lemma for alephfp 10146. (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 6907 | . . 3 ⊢ (𝑣 = ∅ → (𝐻‘𝑣) = (𝐻‘∅)) | |
2 | 1 | eleq1d 2824 | . 2 ⊢ (𝑣 = ∅ → ((𝐻‘𝑣) ∈ ran ℵ ↔ (𝐻‘∅) ∈ ran ℵ)) |
3 | fveq2 6907 | . . 3 ⊢ (𝑣 = 𝑤 → (𝐻‘𝑣) = (𝐻‘𝑤)) | |
4 | 3 | eleq1d 2824 | . 2 ⊢ (𝑣 = 𝑤 → ((𝐻‘𝑣) ∈ ran ℵ ↔ (𝐻‘𝑤) ∈ ran ℵ)) |
5 | fveq2 6907 | . . 3 ⊢ (𝑣 = suc 𝑤 → (𝐻‘𝑣) = (𝐻‘suc 𝑤)) | |
6 | 5 | eleq1d 2824 | . 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 10142 | . 2 ⊢ (𝐻‘∅) ∈ ran ℵ |
9 | alephfnon 10103 | . . . 4 ⊢ ℵ Fn On | |
10 | alephsson 10138 | . . . . 5 ⊢ ran ℵ ⊆ On | |
11 | 10 | sseli 3991 | . . . 4 ⊢ ((𝐻‘𝑤) ∈ ran ℵ → (𝐻‘𝑤) ∈ On) |
12 | fnfvelrn 7100 | . . . 4 ⊢ ((ℵ Fn On ∧ (𝐻‘𝑤) ∈ On) → (ℵ‘(𝐻‘𝑤)) ∈ ran ℵ) | |
13 | 9, 11, 12 | sylancr 587 | . . 3 ⊢ ((𝐻‘𝑤) ∈ ran ℵ → (ℵ‘(𝐻‘𝑤)) ∈ ran ℵ) |
14 | 7 | alephfplem2 10143 | . . . 4 ⊢ (𝑤 ∈ ω → (𝐻‘suc 𝑤) = (ℵ‘(𝐻‘𝑤))) |
15 | 14 | eleq1d 2824 | . . 3 ⊢ (𝑤 ∈ ω → ((𝐻‘suc 𝑤) ∈ ran ℵ ↔ (ℵ‘(𝐻‘𝑤)) ∈ ran ℵ)) |
16 | 13, 15 | imbitrrid 246 | . 2 ⊢ (𝑤 ∈ ω → ((𝐻‘𝑤) ∈ ran ℵ → (𝐻‘suc 𝑤) ∈ ran ℵ)) |
17 | 2, 4, 6, 8, 16 | finds1 7922 | 1 ⊢ (𝑣 ∈ ω → (𝐻‘𝑣) ∈ ran ℵ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∅c0 4339 ran crn 5690 ↾ cres 5691 Oncon0 6386 suc csuc 6388 Fn wfn 6558 ‘cfv 6563 ωcom 7887 reccrdg 8448 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-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-oi 9548 df-har 9595 df-card 9977 df-aleph 9978 |
This theorem is referenced by: alephfplem4 10145 alephfp 10146 |
Copyright terms: Public domain | W3C validator |