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Mirrors > Home > ILE Home > Th. List > infm | GIF version |
Description: An infinite set is inhabited. (Contributed by Jim Kingdon, 18-Feb-2022.) |
Ref | Expression |
---|---|
infm | ⊢ (ω ≼ 𝐴 → ∃𝑥 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brdomi 6636 | . 2 ⊢ (ω ≼ 𝐴 → ∃𝑓 𝑓:ω–1-1→𝐴) | |
2 | f1f 5323 | . . . . 5 ⊢ (𝑓:ω–1-1→𝐴 → 𝑓:ω⟶𝐴) | |
3 | 2 | adantl 275 | . . . 4 ⊢ ((ω ≼ 𝐴 ∧ 𝑓:ω–1-1→𝐴) → 𝑓:ω⟶𝐴) |
4 | peano1 4503 | . . . . 5 ⊢ ∅ ∈ ω | |
5 | 4 | a1i 9 | . . . 4 ⊢ ((ω ≼ 𝐴 ∧ 𝑓:ω–1-1→𝐴) → ∅ ∈ ω) |
6 | 3, 5 | ffvelrnd 5549 | . . 3 ⊢ ((ω ≼ 𝐴 ∧ 𝑓:ω–1-1→𝐴) → (𝑓‘∅) ∈ 𝐴) |
7 | elex2 2697 | . . 3 ⊢ ((𝑓‘∅) ∈ 𝐴 → ∃𝑥 𝑥 ∈ 𝐴) | |
8 | 6, 7 | syl 14 | . 2 ⊢ ((ω ≼ 𝐴 ∧ 𝑓:ω–1-1→𝐴) → ∃𝑥 𝑥 ∈ 𝐴) |
9 | 1, 8 | exlimddv 1870 | 1 ⊢ (ω ≼ 𝐴 → ∃𝑥 𝑥 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∃wex 1468 ∈ wcel 1480 ∅c0 3358 class class class wbr 3924 ωcom 4499 ⟶wf 5114 –1-1→wf1 5115 ‘cfv 5118 ≼ cdom 6626 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-id 4210 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fv 5126 df-dom 6629 |
This theorem is referenced by: infn0 6792 inffiexmid 6793 inffinp1 11931 |
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