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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 9031 | . 2 ⊢ (∃!𝑥 ∈ V 𝜑 ↔ {𝑥 ∈ V ∣ 𝜑} ≈ 1o) | |
2 | reuv 3500 | . 2 ⊢ (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑) | |
3 | rabab 3502 | . . 3 ⊢ {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ 𝜑} | |
4 | 3 | breq1i 5155 | . 2 ⊢ ({𝑥 ∈ V ∣ 𝜑} ≈ 1o ↔ {𝑥 ∣ 𝜑} ≈ 1o) |
5 | 1, 2, 4 | 3bitr3i 301 | 1 ⊢ (∃!𝑥𝜑 ↔ {𝑥 ∣ 𝜑} ≈ 1o) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∃!weu 2561 {cab 2708 ∃!wreu 3373 {crab 3431 Vcvv 3473 class class class wbr 5148 1oc1o 8465 ≈ cen 8942 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 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 8472 df-en 8946 |
This theorem is referenced by: euen1b 9033 modom 9250 |
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