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Mirrors > Home > MPE Home > Th. List > en1top | Structured version Visualization version GIF version |
Description: {∅} is the only topology with one element. (Contributed by FL, 18-Aug-2008.) |
Ref | Expression |
---|---|
en1top | ⊢ (𝐽 ∈ Top → (𝐽 ≈ 1o ↔ 𝐽 = {∅})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0opn 22269 | . . 3 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | |
2 | en1eqsn 9221 | . . . 4 ⊢ ((∅ ∈ 𝐽 ∧ 𝐽 ≈ 1o) → 𝐽 = {∅}) | |
3 | 2 | ex 414 | . . 3 ⊢ (∅ ∈ 𝐽 → (𝐽 ≈ 1o → 𝐽 = {∅})) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝐽 ∈ Top → (𝐽 ≈ 1o → 𝐽 = {∅})) |
5 | id 22 | . . 3 ⊢ (𝐽 = {∅} → 𝐽 = {∅}) | |
6 | 0ex 5265 | . . . 4 ⊢ ∅ ∈ V | |
7 | 6 | ensn1 8964 | . . 3 ⊢ {∅} ≈ 1o |
8 | 5, 7 | eqbrtrdi 5145 | . 2 ⊢ (𝐽 = {∅} → 𝐽 ≈ 1o) |
9 | 4, 8 | impbid1 224 | 1 ⊢ (𝐽 ∈ Top → (𝐽 ≈ 1o ↔ 𝐽 = {∅})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ∅c0 4283 {csn 4587 class class class wbr 5106 1oc1o 8406 ≈ cen 8883 Topctop 22258 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
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-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 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-pw 4563 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-res 5646 df-ima 5647 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-1o 8413 df-en 8887 df-top 22259 |
This theorem is referenced by: hmph0 23162 |
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