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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 3884 | . . . . 5 ⊢ ∅ ∈ V | |
3 | 1, 2 | f1osn 5166 | . . . 4 ⊢ {〈𝐴, ∅〉}:{𝐴}–1-1-onto→{∅} |
4 | 1, 2 | opex 3966 | . . . . . 6 ⊢ 〈𝐴, ∅〉 ∈ V |
5 | 4 | snex 3937 | . . . . 5 ⊢ {〈𝐴, ∅〉} ∈ V |
6 | f1oeq1 5117 | . . . . 5 ⊢ (𝑓 = {〈𝐴, ∅〉} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {〈𝐴, ∅〉}:{𝐴}–1-1-onto→{∅})) | |
7 | 5, 6 | spcev 2647 | . . . 4 ⊢ ({〈𝐴, ∅〉}:{𝐴}–1-1-onto→{∅} → ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}) |
8 | 3, 7 | ax-mp 7 | . . 3 ⊢ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅} |
9 | bren 6228 | . . 3 ⊢ ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}) | |
10 | 8, 9 | mpbir 134 | . 2 ⊢ {𝐴} ≈ {∅} |
11 | df1o2 6013 | . 2 ⊢ 1𝑜 = {∅} | |
12 | 10, 11 | breqtrri 3789 | 1 ⊢ {𝐴} ≈ 1𝑜 |
Colors of variables: wff set class |
Syntax hints: ∃wex 1381 ∈ wcel 1393 Vcvv 2557 ∅c0 3224 {csn 3375 〈cop 3378 class class class wbr 3764 –1-1-onto→wf1o 4901 1𝑜c1o 5994 ≈ cen 6219 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-suc 4108 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-1o 6001 df-en 6222 |
This theorem is referenced by: ensn1g 6277 en1 6279 |
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