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Mirrors > Home > MPE Home > Th. List > alephfplem4 | Structured version Visualization version GIF version |
Description: Lemma for alephfp 10142. (Contributed by NM, 5-Nov-2004.) |
Ref | Expression |
---|---|
alephfplem.1 | ⊢ 𝐻 = (rec(ℵ, ω) ↾ ω) |
Ref | Expression |
---|---|
alephfplem4 | ⊢ ∪ (𝐻 “ ω) ∈ ran ℵ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frfnom 8455 | . . . . 5 ⊢ (rec(ℵ, ω) ↾ ω) Fn ω | |
2 | alephfplem.1 | . . . . . 6 ⊢ 𝐻 = (rec(ℵ, ω) ↾ ω) | |
3 | 2 | fneq1i 6647 | . . . . 5 ⊢ (𝐻 Fn ω ↔ (rec(ℵ, ω) ↾ ω) Fn ω) |
4 | 1, 3 | mpbir 230 | . . . 4 ⊢ 𝐻 Fn ω |
5 | 2 | alephfplem3 10140 | . . . . 5 ⊢ (𝑧 ∈ ω → (𝐻‘𝑧) ∈ ran ℵ) |
6 | 5 | rgen 3053 | . . . 4 ⊢ ∀𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ |
7 | ffnfv 7123 | . . . 4 ⊢ (𝐻:ω⟶ran ℵ ↔ (𝐻 Fn ω ∧ ∀𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ)) | |
8 | 4, 6, 7 | mpbir2an 709 | . . 3 ⊢ 𝐻:ω⟶ran ℵ |
9 | ssun2 4172 | . . 3 ⊢ ran ℵ ⊆ (ω ∪ ran ℵ) | |
10 | fss 6734 | . . 3 ⊢ ((𝐻:ω⟶ran ℵ ∧ ran ℵ ⊆ (ω ∪ ran ℵ)) → 𝐻:ω⟶(ω ∪ ran ℵ)) | |
11 | 8, 9, 10 | mp2an 690 | . 2 ⊢ 𝐻:ω⟶(ω ∪ ran ℵ) |
12 | peano1 7890 | . . 3 ⊢ ∅ ∈ ω | |
13 | 2 | alephfplem1 10138 | . . 3 ⊢ (𝐻‘∅) ∈ ran ℵ |
14 | fveq2 6891 | . . . . 5 ⊢ (𝑧 = ∅ → (𝐻‘𝑧) = (𝐻‘∅)) | |
15 | 14 | eleq1d 2811 | . . . 4 ⊢ (𝑧 = ∅ → ((𝐻‘𝑧) ∈ ran ℵ ↔ (𝐻‘∅) ∈ ran ℵ)) |
16 | 15 | rspcev 3608 | . . 3 ⊢ ((∅ ∈ ω ∧ (𝐻‘∅) ∈ ran ℵ) → ∃𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ) |
17 | 12, 13, 16 | mp2an 690 | . 2 ⊢ ∃𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ |
18 | omex 9677 | . . 3 ⊢ ω ∈ V | |
19 | cardinfima 10131 | . . 3 ⊢ (ω ∈ V → ((𝐻:ω⟶(ω ∪ ran ℵ) ∧ ∃𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ) → ∪ (𝐻 “ ω) ∈ ran ℵ)) | |
20 | 18, 19 | ax-mp 5 | . 2 ⊢ ((𝐻:ω⟶(ω ∪ ran ℵ) ∧ ∃𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ) → ∪ (𝐻 “ ω) ∈ ran ℵ) |
21 | 11, 17, 20 | mp2an 690 | 1 ⊢ ∪ (𝐻 “ ω) ∈ ran ℵ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ∃wrex 3060 Vcvv 3463 ∪ cun 3945 ⊆ wss 3947 ∅c0 4323 ∪ cuni 4906 ran crn 5674 ↾ cres 5675 “ cima 5676 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 ωcom 7866 reccrdg 8429 ℵcale 9970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-inf2 9675 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-int 4948 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7370 df-ov 7417 df-om 7867 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-oi 9544 df-har 9591 df-card 9973 df-aleph 9974 |
This theorem is referenced by: alephfp 10142 alephfp2 10143 |
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