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Mirrors > Home > MPE Home > Th. List > alephfplem4 | Structured version Visualization version GIF version |
Description: Lemma for alephfp 10003. (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 8374 | . . . . 5 ⊢ (rec(ℵ, ω) ↾ ω) Fn ω | |
2 | alephfplem.1 | . . . . . 6 ⊢ 𝐻 = (rec(ℵ, ω) ↾ ω) | |
3 | 2 | fneq1i 6597 | . . . . 5 ⊢ (𝐻 Fn ω ↔ (rec(ℵ, ω) ↾ ω) Fn ω) |
4 | 1, 3 | mpbir 230 | . . . 4 ⊢ 𝐻 Fn ω |
5 | 2 | alephfplem3 10001 | . . . . 5 ⊢ (𝑧 ∈ ω → (𝐻‘𝑧) ∈ ran ℵ) |
6 | 5 | rgen 3065 | . . . 4 ⊢ ∀𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ |
7 | ffnfv 7063 | . . . 4 ⊢ (𝐻:ω⟶ran ℵ ↔ (𝐻 Fn ω ∧ ∀𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ)) | |
8 | 4, 6, 7 | mpbir2an 710 | . . 3 ⊢ 𝐻:ω⟶ran ℵ |
9 | ssun2 4132 | . . 3 ⊢ ran ℵ ⊆ (ω ∪ ran ℵ) | |
10 | fss 6683 | . . 3 ⊢ ((𝐻:ω⟶ran ℵ ∧ ran ℵ ⊆ (ω ∪ ran ℵ)) → 𝐻:ω⟶(ω ∪ ran ℵ)) | |
11 | 8, 9, 10 | mp2an 691 | . 2 ⊢ 𝐻:ω⟶(ω ∪ ran ℵ) |
12 | peano1 7818 | . . 3 ⊢ ∅ ∈ ω | |
13 | 2 | alephfplem1 9999 | . . 3 ⊢ (𝐻‘∅) ∈ ran ℵ |
14 | fveq2 6840 | . . . . 5 ⊢ (𝑧 = ∅ → (𝐻‘𝑧) = (𝐻‘∅)) | |
15 | 14 | eleq1d 2823 | . . . 4 ⊢ (𝑧 = ∅ → ((𝐻‘𝑧) ∈ ran ℵ ↔ (𝐻‘∅) ∈ ran ℵ)) |
16 | 15 | rspcev 3580 | . . 3 ⊢ ((∅ ∈ ω ∧ (𝐻‘∅) ∈ ran ℵ) → ∃𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ) |
17 | 12, 13, 16 | mp2an 691 | . 2 ⊢ ∃𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ |
18 | omex 9538 | . . 3 ⊢ ω ∈ V | |
19 | cardinfima 9992 | . . 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 691 | 1 ⊢ ∪ (𝐻 “ ω) ∈ ran ℵ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3063 ∃wrex 3072 Vcvv 3444 ∪ cun 3907 ⊆ wss 3909 ∅c0 4281 ∪ cuni 4864 ran crn 5633 ↾ cres 5634 “ cima 5635 Fn wfn 6489 ⟶wf 6490 ‘cfv 6494 ωcom 7795 reccrdg 8348 ℵcale 9831 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-inf2 9536 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-se 5588 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-isom 6503 df-riota 7308 df-ov 7355 df-om 7796 df-2nd 7915 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-1o 8405 df-er 8607 df-en 8843 df-dom 8844 df-sdom 8845 df-fin 8846 df-oi 9405 df-har 9452 df-card 9834 df-aleph 9835 |
This theorem is referenced by: alephfp 10003 alephfp2 10004 |
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