Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > euen1 | Structured version Visualization version 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 8702 | . 2 ⊢ (∃!𝑥 ∈ V 𝜑 ↔ {𝑥 ∈ V ∣ 𝜑} ≈ 1o) | |
2 | reuv 3434 | . 2 ⊢ (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑) | |
3 | rabab 3436 | . . 3 ⊢ {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ 𝜑} | |
4 | 3 | breq1i 5060 | . 2 ⊢ ({𝑥 ∈ V ∣ 𝜑} ≈ 1o ↔ {𝑥 ∣ 𝜑} ≈ 1o) |
5 | 1, 2, 4 | 3bitr3i 304 | 1 ⊢ (∃!𝑥𝜑 ↔ {𝑥 ∣ 𝜑} ≈ 1o) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∃!weu 2567 {cab 2714 ∃!wreu 3063 {crab 3065 Vcvv 3408 class class class wbr 5053 1oc1o 8195 ≈ cen 8623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-1o 8202 df-en 8627 |
This theorem is referenced by: euen1b 8704 modom 8879 |
Copyright terms: Public domain | W3C validator |