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Mirrors > Home > MPE Home > Th. List > en1eqsnbi | Structured version Visualization version GIF version |
Description: A set containing an element has exactly one element iff it is a singleton. Formerly part of proof for rngen1zr 20344. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.) |
Ref | Expression |
---|---|
en1eqsnbi | ⊢ (𝐴 ∈ 𝐵 → (𝐵 ≈ 1o ↔ 𝐵 = {𝐴})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | en1eqsn 9270 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = {𝐴}) | |
2 | 1 | ex 414 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐵 ≈ 1o → 𝐵 = {𝐴})) |
3 | ensn1g 9015 | . . 3 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ≈ 1o) | |
4 | breq1 5150 | . . 3 ⊢ (𝐵 = {𝐴} → (𝐵 ≈ 1o ↔ {𝐴} ≈ 1o)) | |
5 | 3, 4 | syl5ibrcom 246 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐵 = {𝐴} → 𝐵 ≈ 1o)) |
6 | 2, 5 | impbid 211 | 1 ⊢ (𝐴 ∈ 𝐵 → (𝐵 ≈ 1o ↔ 𝐵 = {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 {csn 4627 class class class wbr 5147 1oc1o 8454 ≈ cen 8932 |
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-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
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 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-1o 8461 df-en 8936 |
This theorem is referenced by: srgen1zr 20030 rngen1zr 20344 rngosn4 36731 |
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