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Mirrors > Home > MPE Home > Th. List > Mathboxes > gonanegoal | Structured version Visualization version GIF version |
Description: The Godel-set for the Sheffer stroke NAND is not equal to the Godel-set of universal quantification. (Contributed by AV, 21-Oct-2023.) |
Ref | Expression |
---|---|
gonanegoal | ⊢ (𝑎⊼𝑔𝑏) ≠ ∀𝑔𝑖𝑢 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1one2o 8252 | . . . 4 ⊢ 1o ≠ 2o | |
2 | 1 | neii 2989 | . . 3 ⊢ ¬ 1o = 2o |
3 | 2 | intnanr 491 | . 2 ⊢ ¬ (1o = 2o ∧ 〈𝑎, 𝑏〉 = 〈𝑖, 𝑢〉) |
4 | gonafv 32710 | . . . . . 6 ⊢ ((𝑎 ∈ V ∧ 𝑏 ∈ V) → (𝑎⊼𝑔𝑏) = 〈1o, 〈𝑎, 𝑏〉〉) | |
5 | 4 | el2v 3448 | . . . . 5 ⊢ (𝑎⊼𝑔𝑏) = 〈1o, 〈𝑎, 𝑏〉〉 |
6 | df-goal 32702 | . . . . 5 ⊢ ∀𝑔𝑖𝑢 = 〈2o, 〈𝑖, 𝑢〉〉 | |
7 | 5, 6 | eqeq12i 2813 | . . . 4 ⊢ ((𝑎⊼𝑔𝑏) = ∀𝑔𝑖𝑢 ↔ 〈1o, 〈𝑎, 𝑏〉〉 = 〈2o, 〈𝑖, 𝑢〉〉) |
8 | 1oex 8093 | . . . . 5 ⊢ 1o ∈ V | |
9 | opex 5321 | . . . . 5 ⊢ 〈𝑎, 𝑏〉 ∈ V | |
10 | 8, 9 | opth 5333 | . . . 4 ⊢ (〈1o, 〈𝑎, 𝑏〉〉 = 〈2o, 〈𝑖, 𝑢〉〉 ↔ (1o = 2o ∧ 〈𝑎, 𝑏〉 = 〈𝑖, 𝑢〉)) |
11 | 7, 10 | bitri 278 | . . 3 ⊢ ((𝑎⊼𝑔𝑏) = ∀𝑔𝑖𝑢 ↔ (1o = 2o ∧ 〈𝑎, 𝑏〉 = 〈𝑖, 𝑢〉)) |
12 | 11 | necon3abii 3033 | . 2 ⊢ ((𝑎⊼𝑔𝑏) ≠ ∀𝑔𝑖𝑢 ↔ ¬ (1o = 2o ∧ 〈𝑎, 𝑏〉 = 〈𝑖, 𝑢〉)) |
13 | 3, 12 | mpbir 234 | 1 ⊢ (𝑎⊼𝑔𝑏) ≠ ∀𝑔𝑖𝑢 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 = wceq 1538 ≠ wne 2987 Vcvv 3441 〈cop 4531 (class class class)co 7135 1oc1o 8078 2oc2o 8079 ⊼𝑔cgna 32694 ∀𝑔cgol 32695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-om 7561 df-1o 8085 df-2o 8086 df-gona 32701 df-goal 32702 |
This theorem is referenced by: gonarlem 32754 gonar 32755 goalrlem 32756 goalr 32757 fmlasucdisj 32759 |
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