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Mirrors > Home > MPE Home > Th. List > 2ndnpr | Structured version Visualization version GIF version |
Description: Value of the second-member function at non-pairs. (Contributed by Thierry Arnoux, 22-Sep-2017.) |
Ref | Expression |
---|---|
2ndnpr | ⊢ (¬ 𝐴 ∈ (V × V) → (2nd ‘𝐴) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2ndval 7691 | . 2 ⊢ (2nd ‘𝐴) = ∪ ran {𝐴} | |
2 | rnsnn0 6064 | . . . . . 6 ⊢ (𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅) | |
3 | 2 | biimpri 230 | . . . . 5 ⊢ (ran {𝐴} ≠ ∅ → 𝐴 ∈ (V × V)) |
4 | 3 | necon1bi 3044 | . . . 4 ⊢ (¬ 𝐴 ∈ (V × V) → ran {𝐴} = ∅) |
5 | 4 | unieqd 4851 | . . 3 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ ran {𝐴} = ∪ ∅) |
6 | uni0 4865 | . . 3 ⊢ ∪ ∅ = ∅ | |
7 | 5, 6 | syl6eq 2872 | . 2 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ ran {𝐴} = ∅) |
8 | 1, 7 | syl5eq 2868 | 1 ⊢ (¬ 𝐴 ∈ (V × V) → (2nd ‘𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 Vcvv 3494 ∅c0 4290 {csn 4566 ∪ cuni 4837 × cxp 5552 ran crn 5555 ‘cfv 6354 2nd c2nd 7687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-iota 6313 df-fun 6356 df-fv 6362 df-2nd 7689 |
This theorem is referenced by: wlkvv 27407 |
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