![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > euen1 | GIF version |
Description: Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.) |
Ref | Expression |
---|---|
euen1 | ⊢ (∃!𝑥𝜑 ↔ {𝑥 ∣ 𝜑} ≈ 1o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reuen1 6794 | . 2 ⊢ (∃!𝑥 ∈ V 𝜑 ↔ {𝑥 ∈ V ∣ 𝜑} ≈ 1o) | |
2 | reuv 2756 | . 2 ⊢ (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑) | |
3 | rabab 2758 | . . 3 ⊢ {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ 𝜑} | |
4 | 3 | breq1i 4007 | . 2 ⊢ ({𝑥 ∈ V ∣ 𝜑} ≈ 1o ↔ {𝑥 ∣ 𝜑} ≈ 1o) |
5 | 1, 2, 4 | 3bitr3i 210 | 1 ⊢ (∃!𝑥𝜑 ↔ {𝑥 ∣ 𝜑} ≈ 1o) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∃!weu 2026 {cab 2163 ∃!wreu 2457 {crab 2459 Vcvv 2737 class class class wbr 4000 1oc1o 6403 ≈ cen 6731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4205 ax-un 4429 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4289 df-suc 4367 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-1o 6410 df-en 6734 |
This theorem is referenced by: euen1b 6796 |
Copyright terms: Public domain | W3C validator |