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Mirrors > Home > MPE Home > Th. List > ensn1 | Structured version Visualization version 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 4718 | . . . . 5 ⊢ ∅ ∈ V | |
3 | 1, 2 | f1osn 6088 | . . . 4 ⊢ {〈𝐴, ∅〉}:{𝐴}–1-1-onto→{∅} |
4 | snex 4835 | . . . . 5 ⊢ {〈𝐴, ∅〉} ∈ V | |
5 | f1oeq1 6040 | . . . . 5 ⊢ (𝑓 = {〈𝐴, ∅〉} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {〈𝐴, ∅〉}:{𝐴}–1-1-onto→{∅})) | |
6 | 4, 5 | spcev 3273 | . . . 4 ⊢ ({〈𝐴, ∅〉}:{𝐴}–1-1-onto→{∅} → ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}) |
7 | 3, 6 | ax-mp 5 | . . 3 ⊢ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅} |
8 | bren 7850 | . . 3 ⊢ ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}) | |
9 | 7, 8 | mpbir 220 | . 2 ⊢ {𝐴} ≈ {∅} |
10 | df1o2 7459 | . 2 ⊢ 1𝑜 = {∅} | |
11 | 9, 10 | breqtrri 4610 | 1 ⊢ {𝐴} ≈ 1𝑜 |
Colors of variables: wff setvar class |
Syntax hints: ∃wex 1695 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 {csn 4125 〈cop 4131 class class class wbr 4583 –1-1-onto→wf1o 5803 1𝑜c1o 7440 ≈ cen 7838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-suc 5646 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-1o 7447 df-en 7842 |
This theorem is referenced by: ensn1g 7907 en1 7909 fodomfi 8124 pm54.43 8709 1nprm 15230 isprm2lem 15232 gex1 17829 sylow2a 17857 0frgp 18015 en1top 20599 en2top 20600 t1conperf 21049 ptcmplem2 21667 xrge0tsms2 22446 sconpi1 30475 |
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