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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 6611 | . 2 ⊢ (ω ≼ 𝐴 → ∃𝑓 𝑓:ω–1-1→𝐴) | |
2 | f1f 5298 | . . . . 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 4478 | . . . . 5 ⊢ ∅ ∈ ω | |
5 | 4 | a1i 9 | . . . 4 ⊢ ((ω ≼ 𝐴 ∧ 𝑓:ω–1-1→𝐴) → ∅ ∈ ω) |
6 | 3, 5 | ffvelrnd 5524 | . . 3 ⊢ ((ω ≼ 𝐴 ∧ 𝑓:ω–1-1→𝐴) → (𝑓‘∅) ∈ 𝐴) |
7 | elex2 2676 | . . 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 1854 | 1 ⊢ (ω ≼ 𝐴 → ∃𝑥 𝑥 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∃wex 1453 ∈ wcel 1465 ∅c0 3333 class class class wbr 3899 ωcom 4474 ⟶wf 5089 –1-1→wf1 5090 ‘cfv 5093 ≼ cdom 6601 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-id 4185 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fv 5101 df-dom 6604 |
This theorem is referenced by: infn0 6767 inffiexmid 6768 inffinp1 11869 |
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