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Mirrors > Home > MPE Home > Th. List > alephfplem4 | Structured version Visualization version GIF version |
Description: Lemma for alephfp 9528. (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 8064 | . . . . 5 ⊢ (rec(ℵ, ω) ↾ ω) Fn ω | |
2 | alephfplem.1 | . . . . . 6 ⊢ 𝐻 = (rec(ℵ, ω) ↾ ω) | |
3 | 2 | fneq1i 6445 | . . . . 5 ⊢ (𝐻 Fn ω ↔ (rec(ℵ, ω) ↾ ω) Fn ω) |
4 | 1, 3 | mpbir 233 | . . . 4 ⊢ 𝐻 Fn ω |
5 | 2 | alephfplem3 9526 | . . . . 5 ⊢ (𝑧 ∈ ω → (𝐻‘𝑧) ∈ ran ℵ) |
6 | 5 | rgen 3148 | . . . 4 ⊢ ∀𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ |
7 | ffnfv 6877 | . . . 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 4149 | . . 3 ⊢ ran ℵ ⊆ (ω ∪ ran ℵ) | |
10 | fss 6522 | . . 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 7595 | . . 3 ⊢ ∅ ∈ ω | |
13 | 2 | alephfplem1 9524 | . . 3 ⊢ (𝐻‘∅) ∈ ran ℵ |
14 | fveq2 6665 | . . . . 5 ⊢ (𝑧 = ∅ → (𝐻‘𝑧) = (𝐻‘∅)) | |
15 | 14 | eleq1d 2897 | . . . 4 ⊢ (𝑧 = ∅ → ((𝐻‘𝑧) ∈ ran ℵ ↔ (𝐻‘∅) ∈ ran ℵ)) |
16 | 15 | rspcev 3623 | . . 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 9100 | . . 3 ⊢ ω ∈ V | |
19 | cardinfima 9517 | . . 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 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 Vcvv 3495 ∪ cun 3934 ⊆ wss 3936 ∅c0 4291 ∪ cuni 4832 ran crn 5551 ↾ cres 5552 “ cima 5553 Fn wfn 6345 ⟶wf 6346 ‘cfv 6350 ωcom 7574 reccrdg 8039 ℵcale 9359 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-oi 8968 df-har 9016 df-card 9362 df-aleph 9363 |
This theorem is referenced by: alephfp 9528 alephfp2 9529 |
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