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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 7673 | . 2 ⊢ (2nd ‘𝐴) = ∪ ran {𝐴} | |
2 | rnsnn0 6046 | . . . . . 6 ⊢ (𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅) | |
3 | 2 | biimpri 230 | . . . . 5 ⊢ (ran {𝐴} ≠ ∅ → 𝐴 ∈ (V × V)) |
4 | 3 | necon1bi 3039 | . . . 4 ⊢ (¬ 𝐴 ∈ (V × V) → ran {𝐴} = ∅) |
5 | 4 | unieqd 4833 | . . 3 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ ran {𝐴} = ∪ ∅) |
6 | uni0 4847 | . . 3 ⊢ ∪ ∅ = ∅ | |
7 | 5, 6 | syl6eq 2871 | . 2 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ ran {𝐴} = ∅) |
8 | 1, 7 | syl5eq 2867 | 1 ⊢ (¬ 𝐴 ∈ (V × V) → (2nd ‘𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2114 ≠ wne 3011 Vcvv 3481 ∅c0 4274 {csn 4548 ∪ cuni 4819 × cxp 5534 ran crn 5537 ‘cfv 6336 2nd c2nd 7669 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-ral 3138 df-rex 3139 df-rab 3142 df-v 3483 df-sbc 3759 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-nul 4275 df-if 4449 df-sn 4549 df-pr 4551 df-op 4555 df-uni 4820 df-br 5048 df-opab 5110 df-mpt 5128 df-id 5441 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-iota 6295 df-fun 6338 df-fv 6344 df-2nd 7671 |
This theorem is referenced by: wlkvv 27389 |
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