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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 7973 | . 2 ⊢ (1st ‘𝐴) = ∪ dom {𝐴} | |
2 | dmsnn0 6199 | . . . . . 6 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅) | |
3 | 2 | biimpri 227 | . . . . 5 ⊢ (dom {𝐴} ≠ ∅ → 𝐴 ∈ (V × V)) |
4 | 3 | necon1bi 2963 | . . . 4 ⊢ (¬ 𝐴 ∈ (V × V) → dom {𝐴} = ∅) |
5 | 4 | unieqd 4915 | . . 3 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ dom {𝐴} = ∪ ∅) |
6 | uni0 4932 | . . 3 ⊢ ∪ ∅ = ∅ | |
7 | 5, 6 | eqtrdi 2782 | . 2 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ dom {𝐴} = ∅) |
8 | 1, 7 | eqtrid 2778 | 1 ⊢ (¬ 𝐴 ∈ (V × V) → (1st ‘𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 Vcvv 3468 ∅c0 4317 {csn 4623 ∪ cuni 4902 × cxp 5667 dom cdm 5669 ‘cfv 6536 1st c1st 7969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6488 df-fun 6538 df-fv 6544 df-1st 7971 |
This theorem is referenced by: (None) |
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