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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 7925 | . 2 ⊢ (2nd ‘𝐴) = ∪ ran {𝐴} | |
2 | rnsnn0 6161 | . . . . . 6 ⊢ (𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅) | |
3 | 2 | biimpri 227 | . . . . 5 ⊢ (ran {𝐴} ≠ ∅ → 𝐴 ∈ (V × V)) |
4 | 3 | necon1bi 2973 | . . . 4 ⊢ (¬ 𝐴 ∈ (V × V) → ran {𝐴} = ∅) |
5 | 4 | unieqd 4880 | . . 3 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ ran {𝐴} = ∪ ∅) |
6 | uni0 4897 | . . 3 ⊢ ∪ ∅ = ∅ | |
7 | 5, 6 | eqtrdi 2793 | . 2 ⊢ (¬ 𝐴 ∈ (V × V) → ∪ ran {𝐴} = ∅) |
8 | 1, 7 | eqtrid 2789 | 1 ⊢ (¬ 𝐴 ∈ (V × V) → (2nd ‘𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 Vcvv 3446 ∅c0 4283 {csn 4587 ∪ cuni 4866 × cxp 5632 ran crn 5635 ‘cfv 6497 2nd c2nd 7921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-iota 6449 df-fun 6499 df-fv 6505 df-2nd 7923 |
This theorem is referenced by: wlkvv 28578 |
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