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Mirrors > Home > ILE Home > Th. List > fict | GIF version |
Description: A finite set is dominated by ω. Also see finct 7175. (Contributed by Thierry Arnoux, 27-Mar-2018.) |
Ref | Expression |
---|---|
fict | ⊢ (𝐴 ∈ Fin → 𝐴 ≼ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfi 6815 | . . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ω 𝐴 ≈ 𝑛) | |
2 | 1 | biimpi 120 | . 2 ⊢ (𝐴 ∈ Fin → ∃𝑛 ∈ ω 𝐴 ≈ 𝑛) |
3 | simprr 531 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → 𝐴 ≈ 𝑛) | |
4 | omex 4625 | . . . . 5 ⊢ ω ∈ V | |
5 | ordom 4639 | . . . . . 6 ⊢ Ord ω | |
6 | ordelss 4410 | . . . . . 6 ⊢ ((Ord ω ∧ 𝑛 ∈ ω) → 𝑛 ⊆ ω) | |
7 | 5, 6 | mpan 424 | . . . . 5 ⊢ (𝑛 ∈ ω → 𝑛 ⊆ ω) |
8 | ssdomg 6832 | . . . . 5 ⊢ (ω ∈ V → (𝑛 ⊆ ω → 𝑛 ≼ ω)) | |
9 | 4, 7, 8 | mpsyl 65 | . . . 4 ⊢ (𝑛 ∈ ω → 𝑛 ≼ ω) |
10 | 9 | ad2antrl 490 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → 𝑛 ≼ ω) |
11 | endomtr 6844 | . . 3 ⊢ ((𝐴 ≈ 𝑛 ∧ 𝑛 ≼ ω) → 𝐴 ≼ ω) | |
12 | 3, 10, 11 | syl2anc 411 | . 2 ⊢ ((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → 𝐴 ≼ ω) |
13 | 2, 12 | rexlimddv 2616 | 1 ⊢ (𝐴 ∈ Fin → 𝐴 ≼ ω) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2164 ∃wrex 2473 Vcvv 2760 ⊆ wss 3153 class class class wbr 4029 Ord word 4393 ωcom 4622 ≈ cen 6792 ≼ cdom 6793 Fincfn 6794 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-iinf 4620 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-br 4030 df-opab 4091 df-tr 4128 df-id 4324 df-iord 4397 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-en 6795 df-dom 6796 df-fin 6797 |
This theorem is referenced by: (None) |
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