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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 8016 | . 2 ⊢ (1st ‘𝐴) = ∪ dom {𝐴} | |
| 2 | dmsnn0 6227 | . . . . . 6 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅) | |
| 3 | 2 | biimpri 228 | . . . . 5 ⊢ (dom {𝐴} ≠ ∅ → 𝐴 ∈ (V × V)) |
| 4 | 3 | necon1bi 2969 | . . . 4 ⊢ (¬ 𝐴 ∈ (V × V) → dom {𝐴} = ∅) |
| 5 | 4 | unieqd 4920 | . . 3 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ dom {𝐴} = ∪ ∅) |
| 6 | uni0 4935 | . . 3 ⊢ ∪ ∅ = ∅ | |
| 7 | 5, 6 | eqtrdi 2793 | . 2 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ dom {𝐴} = ∅) |
| 8 | 1, 7 | eqtrid 2789 | 1 ⊢ (¬ 𝐴 ∈ (V × V) → (1st ‘𝐴) = ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∅c0 4333 {csn 4626 ∪ cuni 4907 × cxp 5683 dom cdm 5685 ‘cfv 6561 1st c1st 8012 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-1st 8014 |
| This theorem is referenced by: (None) |
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