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Mirrors > Home > MPE Home > Th. List > ensn1OLD | Structured version Visualization version GIF version |
Description: Obsolete version of ensn1 8964 as of 23-Sep-2024. (Contributed by NM, 4-Nov-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ensn1OLD.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
ensn1OLD | ⊢ {𝐴} ≈ 1o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex 5389 | . . . 4 ⊢ {⟨𝐴, ∅⟩} ∈ V | |
2 | f1oeq1 6773 | . . . 4 ⊢ (𝑓 = {⟨𝐴, ∅⟩} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅})) | |
3 | ensn1OLD.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
4 | 0ex 5265 | . . . . 5 ⊢ ∅ ∈ V | |
5 | 3, 4 | f1osn 6825 | . . . 4 ⊢ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅} |
6 | 1, 2, 5 | ceqsexv2d 3496 | . . 3 ⊢ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅} |
7 | bren 8896 | . . 3 ⊢ ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}) | |
8 | 6, 7 | mpbir 230 | . 2 ⊢ {𝐴} ≈ {∅} |
9 | df1o2 8420 | . 2 ⊢ 1o = {∅} | |
10 | 8, 9 | breqtrri 5133 | 1 ⊢ {𝐴} ≈ 1o |
Colors of variables: wff setvar class |
Syntax hints: ∃wex 1782 ∈ wcel 2107 Vcvv 3444 ∅c0 4283 {csn 4587 ⟨cop 4593 class class class wbr 5106 –1-1-onto→wf1o 6496 1oc1o 8406 ≈ cen 8883 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-mo 2535 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-suc 6324 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-1o 8413 df-en 8887 |
This theorem is referenced by: (None) |
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