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Mirrors > Home > ILE Home > Th. List > ensn1 | GIF version |
Description: A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.) |
Ref | Expression |
---|---|
ensn1.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
ensn1 | ⊢ {𝐴} ≈ 1o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ensn1.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
2 | 0ex 4055 | . . . . 5 ⊢ ∅ ∈ V | |
3 | 1, 2 | f1osn 5407 | . . . 4 ⊢ {〈𝐴, ∅〉}:{𝐴}–1-1-onto→{∅} |
4 | 1, 2 | opex 4151 | . . . . . 6 ⊢ 〈𝐴, ∅〉 ∈ V |
5 | 4 | snex 4109 | . . . . 5 ⊢ {〈𝐴, ∅〉} ∈ V |
6 | f1oeq1 5356 | . . . . 5 ⊢ (𝑓 = {〈𝐴, ∅〉} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {〈𝐴, ∅〉}:{𝐴}–1-1-onto→{∅})) | |
7 | 5, 6 | spcev 2780 | . . . 4 ⊢ ({〈𝐴, ∅〉}:{𝐴}–1-1-onto→{∅} → ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}) |
8 | 3, 7 | ax-mp 5 | . . 3 ⊢ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅} |
9 | bren 6641 | . . 3 ⊢ ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}) | |
10 | 8, 9 | mpbir 145 | . 2 ⊢ {𝐴} ≈ {∅} |
11 | df1o2 6326 | . 2 ⊢ 1o = {∅} | |
12 | 10, 11 | breqtrri 3955 | 1 ⊢ {𝐴} ≈ 1o |
Colors of variables: wff set class |
Syntax hints: ∃wex 1468 ∈ wcel 1480 Vcvv 2686 ∅c0 3363 {csn 3527 〈cop 3530 class class class wbr 3929 –1-1-onto→wf1o 5122 1oc1o 6306 ≈ cen 6632 |
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 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-suc 4293 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-1o 6313 df-en 6635 |
This theorem is referenced by: ensn1g 6691 en1 6693 pm54.43 7046 1nprm 11795 en1top 12246 |
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