Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > fict | GIF version |
Description: A finite set is dominated by ω. Also see finct 7072. (Contributed by Thierry Arnoux, 27-Mar-2018.) |
Ref | Expression |
---|---|
fict | ⊢ (𝐴 ∈ Fin → 𝐴 ≼ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfi 6718 | . . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ω 𝐴 ≈ 𝑛) | |
2 | 1 | biimpi 119 | . 2 ⊢ (𝐴 ∈ Fin → ∃𝑛 ∈ ω 𝐴 ≈ 𝑛) |
3 | simprr 522 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → 𝐴 ≈ 𝑛) | |
4 | omex 4564 | . . . . 5 ⊢ ω ∈ V | |
5 | ordom 4578 | . . . . . 6 ⊢ Ord ω | |
6 | ordelss 4351 | . . . . . 6 ⊢ ((Ord ω ∧ 𝑛 ∈ ω) → 𝑛 ⊆ ω) | |
7 | 5, 6 | mpan 421 | . . . . 5 ⊢ (𝑛 ∈ ω → 𝑛 ⊆ ω) |
8 | ssdomg 6735 | . . . . 5 ⊢ (ω ∈ V → (𝑛 ⊆ ω → 𝑛 ≼ ω)) | |
9 | 4, 7, 8 | mpsyl 65 | . . . 4 ⊢ (𝑛 ∈ ω → 𝑛 ≼ ω) |
10 | 9 | ad2antrl 482 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → 𝑛 ≼ ω) |
11 | endomtr 6747 | . . 3 ⊢ ((𝐴 ≈ 𝑛 ∧ 𝑛 ≼ ω) → 𝐴 ≼ ω) | |
12 | 3, 10, 11 | syl2anc 409 | . 2 ⊢ ((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → 𝐴 ≼ ω) |
13 | 2, 12 | rexlimddv 2586 | 1 ⊢ (𝐴 ∈ Fin → 𝐴 ≼ ω) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2135 ∃wrex 2443 Vcvv 2721 ⊆ wss 3111 class class class wbr 3976 Ord word 4334 ωcom 4561 ≈ cen 6695 ≼ cdom 6696 Fincfn 6697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-tr 4075 df-id 4265 df-iord 4338 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-en 6698 df-dom 6699 df-fin 6700 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |