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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 7995 | . 2 ⊢ (1st ‘𝐴) = ∪ dom {𝐴} | |
2 | dmsnn0 6211 | . . . . . 6 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅) | |
3 | 2 | biimpri 227 | . . . . 5 ⊢ (dom {𝐴} ≠ ∅ → 𝐴 ∈ (V × V)) |
4 | 3 | necon1bi 2966 | . . . 4 ⊢ (¬ 𝐴 ∈ (V × V) → dom {𝐴} = ∅) |
5 | 4 | unieqd 4921 | . . 3 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ dom {𝐴} = ∪ ∅) |
6 | uni0 4938 | . . 3 ⊢ ∪ ∅ = ∅ | |
7 | 5, 6 | eqtrdi 2784 | . 2 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ dom {𝐴} = ∅) |
8 | 1, 7 | eqtrid 2780 | 1 ⊢ (¬ 𝐴 ∈ (V × V) → (1st ‘𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 Vcvv 3471 ∅c0 4323 {csn 4629 ∪ cuni 4908 × cxp 5676 dom cdm 5678 ‘cfv 6548 1st c1st 7991 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-iota 6500 df-fun 6550 df-fv 6556 df-1st 7993 |
This theorem is referenced by: (None) |
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