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Mirrors > Home > MPE Home > Th. List > alephfplem4 | Structured version Visualization version GIF version |
Description: Lemma for alephfp 9217. (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 7769 | . . . . 5 ⊢ (rec(ℵ, ω) ↾ ω) Fn ω | |
2 | alephfplem.1 | . . . . . 6 ⊢ 𝐻 = (rec(ℵ, ω) ↾ ω) | |
3 | 2 | fneq1i 6196 | . . . . 5 ⊢ (𝐻 Fn ω ↔ (rec(ℵ, ω) ↾ ω) Fn ω) |
4 | 1, 3 | mpbir 223 | . . . 4 ⊢ 𝐻 Fn ω |
5 | 2 | alephfplem3 9215 | . . . . 5 ⊢ (𝑧 ∈ ω → (𝐻‘𝑧) ∈ ran ℵ) |
6 | 5 | rgen 3103 | . . . 4 ⊢ ∀𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ |
7 | ffnfv 6614 | . . . 4 ⊢ (𝐻:ω⟶ran ℵ ↔ (𝐻 Fn ω ∧ ∀𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ)) | |
8 | 4, 6, 7 | mpbir2an 703 | . . 3 ⊢ 𝐻:ω⟶ran ℵ |
9 | ssun2 3975 | . . 3 ⊢ ran ℵ ⊆ (ω ∪ ran ℵ) | |
10 | fss 6269 | . . 3 ⊢ ((𝐻:ω⟶ran ℵ ∧ ran ℵ ⊆ (ω ∪ ran ℵ)) → 𝐻:ω⟶(ω ∪ ran ℵ)) | |
11 | 8, 9, 10 | mp2an 684 | . 2 ⊢ 𝐻:ω⟶(ω ∪ ran ℵ) |
12 | peano1 7319 | . . 3 ⊢ ∅ ∈ ω | |
13 | 2 | alephfplem1 9213 | . . 3 ⊢ (𝐻‘∅) ∈ ran ℵ |
14 | fveq2 6411 | . . . . 5 ⊢ (𝑧 = ∅ → (𝐻‘𝑧) = (𝐻‘∅)) | |
15 | 14 | eleq1d 2863 | . . . 4 ⊢ (𝑧 = ∅ → ((𝐻‘𝑧) ∈ ran ℵ ↔ (𝐻‘∅) ∈ ran ℵ)) |
16 | 15 | rspcev 3497 | . . 3 ⊢ ((∅ ∈ ω ∧ (𝐻‘∅) ∈ ran ℵ) → ∃𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ) |
17 | 12, 13, 16 | mp2an 684 | . 2 ⊢ ∃𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ |
18 | omex 8790 | . . 3 ⊢ ω ∈ V | |
19 | cardinfima 9206 | . . 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 684 | 1 ⊢ ∪ (𝐻 “ ω) ∈ ran ℵ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∀wral 3089 ∃wrex 3090 Vcvv 3385 ∪ cun 3767 ⊆ wss 3769 ∅c0 4115 ∪ cuni 4628 ran crn 5313 ↾ cres 5314 “ cima 5315 Fn wfn 6096 ⟶wf 6097 ‘cfv 6101 ωcom 7299 reccrdg 7744 ℵcale 9048 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-inf2 8788 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-se 5272 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-isom 6110 df-riota 6839 df-om 7300 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-oi 8657 df-har 8705 df-card 9051 df-aleph 9052 |
This theorem is referenced by: alephfp 9217 alephfp2 9218 |
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