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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 4025 | . . . . 5 ⊢ ∅ ∈ V | |
3 | 1, 2 | f1osn 5375 | . . . 4 ⊢ {〈𝐴, ∅〉}:{𝐴}–1-1-onto→{∅} |
4 | 1, 2 | opex 4121 | . . . . . 6 ⊢ 〈𝐴, ∅〉 ∈ V |
5 | 4 | snex 4079 | . . . . 5 ⊢ {〈𝐴, ∅〉} ∈ V |
6 | f1oeq1 5326 | . . . . 5 ⊢ (𝑓 = {〈𝐴, ∅〉} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {〈𝐴, ∅〉}:{𝐴}–1-1-onto→{∅})) | |
7 | 5, 6 | spcev 2754 | . . . 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 6609 | . . 3 ⊢ ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}) | |
10 | 8, 9 | mpbir 145 | . 2 ⊢ {𝐴} ≈ {∅} |
11 | df1o2 6294 | . 2 ⊢ 1o = {∅} | |
12 | 10, 11 | breqtrri 3925 | 1 ⊢ {𝐴} ≈ 1o |
Colors of variables: wff set class |
Syntax hints: ∃wex 1453 ∈ wcel 1465 Vcvv 2660 ∅c0 3333 {csn 3497 〈cop 3500 class class class wbr 3899 –1-1-onto→wf1o 5092 1oc1o 6274 ≈ cen 6600 |
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-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-br 3900 df-opab 3960 df-id 4185 df-suc 4263 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-1o 6281 df-en 6603 |
This theorem is referenced by: ensn1g 6659 en1 6661 pm54.43 7014 1nprm 11722 en1top 12173 |
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