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Mirrors > Home > ILE Home > Th. List > ctex | GIF version |
Description: A countable set is a set. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
Ref | Expression |
---|---|
ctex | ⊢ (𝐴 ≼ ω → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brdomi 6396 | . 2 ⊢ (𝐴 ≼ ω → ∃𝑓 𝑓:𝐴–1-1→ω) | |
2 | f1dm 5169 | . . . 4 ⊢ (𝑓:𝐴–1-1→ω → dom 𝑓 = 𝐴) | |
3 | vex 2615 | . . . . 5 ⊢ 𝑓 ∈ V | |
4 | 3 | dmex 4657 | . . . 4 ⊢ dom 𝑓 ∈ V |
5 | 2, 4 | syl6eqelr 2174 | . . 3 ⊢ (𝑓:𝐴–1-1→ω → 𝐴 ∈ V) |
6 | 5 | exlimiv 1530 | . 2 ⊢ (∃𝑓 𝑓:𝐴–1-1→ω → 𝐴 ∈ V) |
7 | 1, 6 | syl 14 | 1 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∃wex 1422 ∈ wcel 1434 Vcvv 2612 class class class wbr 3811 ωcom 4368 dom cdm 4401 –1-1→wf1 4966 ≼ cdom 6386 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 4000 ax-un 4224 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-v 2614 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-br 3812 df-opab 3866 df-xp 4407 df-rel 4408 df-cnv 4409 df-dm 4411 df-rn 4412 df-fn 4972 df-f 4973 df-f1 4974 df-dom 6389 |
This theorem is referenced by: cnvct 6456 ssct 6464 xpct 10989 |
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