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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 20756. (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 9308 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = {𝐴}) | |
2 | 1 | ex 411 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐵 ≈ 1o → 𝐵 = {𝐴})) |
3 | ensn1g 9055 | . . 3 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ≈ 1o) | |
4 | breq1 5156 | . . 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 1534 ∈ wcel 2099 {csn 4633 class class class wbr 5153 1oc1o 8489 ≈ cen 8971 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-1o 8496 df-en 8975 |
This theorem is referenced by: srgen1zr 20199 rngen1zr 20756 rngosn4 37626 |
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