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Mirrors > Home > MPE Home > Th. List > eusv2 | Structured version Visualization version GIF version |
Description: Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by NM, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) |
Ref | Expression |
---|---|
eusv2.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
eusv2 | ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 ↔ ∃!𝑦∀𝑥 𝑦 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eusv2.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | 1 | eusv2nf 5288 | . 2 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 ↔ Ⅎ𝑥𝐴) |
3 | eusvnfb 5286 | . . 3 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ (Ⅎ𝑥𝐴 ∧ 𝐴 ∈ V)) | |
4 | 1, 3 | mpbiran2 710 | . 2 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 ↔ Ⅎ𝑥𝐴) |
5 | 2, 4 | bitr4i 281 | 1 ⊢ (∃!𝑦∃𝑥 𝑦 = 𝐴 ↔ ∃!𝑦∀𝑥 𝑦 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∀wal 1541 = wceq 1543 ∃wex 1787 ∈ wcel 2110 ∃!weu 2567 Ⅎwnfc 2884 Vcvv 3408 |
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 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 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-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-sn 4542 df-pr 4544 df-uni 4820 |
This theorem is referenced by: (None) |
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