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Mirrors > Home > MPE Home > Th. List > alephfplem4 | Structured version Visualization version GIF version |
Description: Lemma for alephfp 9387. (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 7929 | . . . . 5 ⊢ (rec(ℵ, ω) ↾ ω) Fn ω | |
2 | alephfplem.1 | . . . . . 6 ⊢ 𝐻 = (rec(ℵ, ω) ↾ ω) | |
3 | 2 | fneq1i 6327 | . . . . 5 ⊢ (𝐻 Fn ω ↔ (rec(ℵ, ω) ↾ ω) Fn ω) |
4 | 1, 3 | mpbir 232 | . . . 4 ⊢ 𝐻 Fn ω |
5 | 2 | alephfplem3 9385 | . . . . 5 ⊢ (𝑧 ∈ ω → (𝐻‘𝑧) ∈ ran ℵ) |
6 | 5 | rgen 3117 | . . . 4 ⊢ ∀𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ |
7 | ffnfv 6752 | . . . 4 ⊢ (𝐻:ω⟶ran ℵ ↔ (𝐻 Fn ω ∧ ∀𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ)) | |
8 | 4, 6, 7 | mpbir2an 707 | . . 3 ⊢ 𝐻:ω⟶ran ℵ |
9 | ssun2 4076 | . . 3 ⊢ ran ℵ ⊆ (ω ∪ ran ℵ) | |
10 | fss 6402 | . . 3 ⊢ ((𝐻:ω⟶ran ℵ ∧ ran ℵ ⊆ (ω ∪ ran ℵ)) → 𝐻:ω⟶(ω ∪ ran ℵ)) | |
11 | 8, 9, 10 | mp2an 688 | . 2 ⊢ 𝐻:ω⟶(ω ∪ ran ℵ) |
12 | peano1 7464 | . . 3 ⊢ ∅ ∈ ω | |
13 | 2 | alephfplem1 9383 | . . 3 ⊢ (𝐻‘∅) ∈ ran ℵ |
14 | fveq2 6545 | . . . . 5 ⊢ (𝑧 = ∅ → (𝐻‘𝑧) = (𝐻‘∅)) | |
15 | 14 | eleq1d 2869 | . . . 4 ⊢ (𝑧 = ∅ → ((𝐻‘𝑧) ∈ ran ℵ ↔ (𝐻‘∅) ∈ ran ℵ)) |
16 | 15 | rspcev 3561 | . . 3 ⊢ ((∅ ∈ ω ∧ (𝐻‘∅) ∈ ran ℵ) → ∃𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ) |
17 | 12, 13, 16 | mp2an 688 | . 2 ⊢ ∃𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ |
18 | omex 8959 | . . 3 ⊢ ω ∈ V | |
19 | cardinfima 9376 | . . 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 688 | 1 ⊢ ∪ (𝐻 “ ω) ∈ ran ℵ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1525 ∈ wcel 2083 ∀wral 3107 ∃wrex 3108 Vcvv 3440 ∪ cun 3863 ⊆ wss 3865 ∅c0 4217 ∪ cuni 4751 ran crn 5451 ↾ cres 5452 “ cima 5453 Fn wfn 6227 ⟶wf 6228 ‘cfv 6232 ωcom 7443 reccrdg 7904 ℵcale 9218 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-inf2 8957 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-se 5410 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-isom 6241 df-riota 6984 df-om 7444 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-oi 8827 df-har 8875 df-card 9221 df-aleph 9222 |
This theorem is referenced by: alephfp 9387 alephfp2 9388 |
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