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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 8564 | . 2 ⊢ (∃!𝑥 ∈ V 𝜑 ↔ {𝑥 ∈ V ∣ 𝜑} ≈ 1o) | |
2 | reuv 3513 | . 2 ⊢ (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑) | |
3 | rabab 3515 | . . 3 ⊢ {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ 𝜑} | |
4 | 3 | breq1i 5059 | . 2 ⊢ ({𝑥 ∈ V ∣ 𝜑} ≈ 1o ↔ {𝑥 ∣ 𝜑} ≈ 1o) |
5 | 1, 2, 4 | 3bitr3i 303 | 1 ⊢ (∃!𝑥𝜑 ↔ {𝑥 ∣ 𝜑} ≈ 1o) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∃!weu 2653 {cab 2799 ∃!wreu 3140 {crab 3142 Vcvv 3486 class class class wbr 5052 1oc1o 8081 ≈ cen 8492 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pr 5316 ax-un 7447 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3488 df-sbc 3764 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-opab 5115 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-1o 8088 df-en 8496 |
This theorem is referenced by: euen1b 8566 modom 8705 |
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