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Mirrors > Home > MPE Home > Th. List > 1stnpr | Structured version Visualization version GIF version |
Description: Value of the first-member function at non-pairs. (Contributed by Thierry Arnoux, 22-Sep-2017.) |
Ref | Expression |
---|---|
1stnpr | ⊢ (¬ 𝐴 ∈ (V × V) → (1st ‘𝐴) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1stval 7865 | . 2 ⊢ (1st ‘𝐴) = ∪ dom {𝐴} | |
2 | dmsnn0 6125 | . . . . . 6 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅) | |
3 | 2 | biimpri 227 | . . . . 5 ⊢ (dom {𝐴} ≠ ∅ → 𝐴 ∈ (V × V)) |
4 | 3 | necon1bi 2970 | . . . 4 ⊢ (¬ 𝐴 ∈ (V × V) → dom {𝐴} = ∅) |
5 | 4 | unieqd 4858 | . . 3 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ dom {𝐴} = ∪ ∅) |
6 | uni0 4875 | . . 3 ⊢ ∪ ∅ = ∅ | |
7 | 5, 6 | eqtrdi 2792 | . 2 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ dom {𝐴} = ∅) |
8 | 1, 7 | eqtrid 2788 | 1 ⊢ (¬ 𝐴 ∈ (V × V) → (1st ‘𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1539 ∈ wcel 2104 ≠ wne 2941 Vcvv 3437 ∅c0 4262 {csn 4565 ∪ cuni 4844 × cxp 5598 dom cdm 5600 ‘cfv 6458 1st c1st 7861 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pr 5361 ax-un 7620 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3287 df-v 3439 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-br 5082 df-opab 5144 df-mpt 5165 df-id 5500 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-iota 6410 df-fun 6460 df-fv 6466 df-1st 7863 |
This theorem is referenced by: (None) |
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