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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 8269 | . . . 4 ⊢ 1o ≠ 2o | |
2 | 1 | neii 3018 | . . 3 ⊢ ¬ 1o = 2o |
3 | 2 | intnanr 490 | . 2 ⊢ ¬ (1o = 2o ∧ 〈𝑎, 𝑏〉 = 〈𝑖, 𝑢〉) |
4 | gonafv 32597 | . . . . . 6 ⊢ ((𝑎 ∈ V ∧ 𝑏 ∈ V) → (𝑎⊼𝑔𝑏) = 〈1o, 〈𝑎, 𝑏〉〉) | |
5 | 4 | el2v 3501 | . . . . 5 ⊢ (𝑎⊼𝑔𝑏) = 〈1o, 〈𝑎, 𝑏〉〉 |
6 | df-goal 32589 | . . . . 5 ⊢ ∀𝑔𝑖𝑢 = 〈2o, 〈𝑖, 𝑢〉〉 | |
7 | 5, 6 | eqeq12i 2836 | . . . 4 ⊢ ((𝑎⊼𝑔𝑏) = ∀𝑔𝑖𝑢 ↔ 〈1o, 〈𝑎, 𝑏〉〉 = 〈2o, 〈𝑖, 𝑢〉〉) |
8 | 1oex 8110 | . . . . 5 ⊢ 1o ∈ V | |
9 | opex 5356 | . . . . 5 ⊢ 〈𝑎, 𝑏〉 ∈ V | |
10 | 8, 9 | opth 5368 | . . . 4 ⊢ (〈1o, 〈𝑎, 𝑏〉〉 = 〈2o, 〈𝑖, 𝑢〉〉 ↔ (1o = 2o ∧ 〈𝑎, 𝑏〉 = 〈𝑖, 𝑢〉)) |
11 | 7, 10 | bitri 277 | . . 3 ⊢ ((𝑎⊼𝑔𝑏) = ∀𝑔𝑖𝑢 ↔ (1o = 2o ∧ 〈𝑎, 𝑏〉 = 〈𝑖, 𝑢〉)) |
12 | 11 | necon3abii 3062 | . 2 ⊢ ((𝑎⊼𝑔𝑏) ≠ ∀𝑔𝑖𝑢 ↔ ¬ (1o = 2o ∧ 〈𝑎, 𝑏〉 = 〈𝑖, 𝑢〉)) |
13 | 3, 12 | mpbir 233 | 1 ⊢ (𝑎⊼𝑔𝑏) ≠ ∀𝑔𝑖𝑢 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 = wceq 1537 ≠ wne 3016 Vcvv 3494 〈cop 4573 (class class class)co 7156 1oc1o 8095 2oc2o 8096 ⊼𝑔cgna 32581 ∀𝑔cgol 32582 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-om 7581 df-1o 8102 df-2o 8103 df-gona 32588 df-goal 32589 |
This theorem is referenced by: gonarlem 32641 gonar 32642 goalrlem 32643 goalr 32644 fmlasucdisj 32646 |
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