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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 7680 | . 2 ⊢ (1st ‘𝐴) = ∪ dom {𝐴} | |
2 | dmsnn0 6057 | . . . . . 6 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅) | |
3 | 2 | biimpri 229 | . . . . 5 ⊢ (dom {𝐴} ≠ ∅ → 𝐴 ∈ (V × V)) |
4 | 3 | necon1bi 3041 | . . . 4 ⊢ (¬ 𝐴 ∈ (V × V) → dom {𝐴} = ∅) |
5 | 4 | unieqd 4840 | . . 3 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ dom {𝐴} = ∪ ∅) |
6 | uni0 4857 | . . 3 ⊢ ∪ ∅ = ∅ | |
7 | 5, 6 | syl6eq 2869 | . 2 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ dom {𝐴} = ∅) |
8 | 1, 7 | syl5eq 2865 | 1 ⊢ (¬ 𝐴 ∈ (V × V) → (1st ‘𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 Vcvv 3492 ∅c0 4288 {csn 4557 ∪ cuni 4830 × cxp 5546 dom cdm 5548 ‘cfv 6348 1st c1st 7676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fv 6356 df-1st 7678 |
This theorem is referenced by: (None) |
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