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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 20397. (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 9273 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = {𝐴}) | |
2 | 1 | ex 413 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐵 ≈ 1o → 𝐵 = {𝐴})) |
3 | ensn1g 9018 | . . 3 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ≈ 1o) | |
4 | breq1 5151 | . . 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 1541 ∈ wcel 2106 {csn 4628 class class class wbr 5148 1oc1o 8458 ≈ cen 8935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-1o 8465 df-en 8939 |
This theorem is referenced by: srgen1zr 20038 rngen1zr 20397 rngosn4 36788 |
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