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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 8262 | . . . 4 ⊢ 1o ≠ 2o | |
2 | 1 | neii 3017 | . . 3 ⊢ ¬ 1o = 2o |
3 | 2 | intnanr 490 | . 2 ⊢ ¬ (1o = 2o ∧ 〈𝑎, 𝑏〉 = 〈𝑖, 𝑢〉) |
4 | gonafv 32618 | . . . . . 6 ⊢ ((𝑎 ∈ V ∧ 𝑏 ∈ V) → (𝑎⊼𝑔𝑏) = 〈1o, 〈𝑎, 𝑏〉〉) | |
5 | 4 | el2v 3498 | . . . . 5 ⊢ (𝑎⊼𝑔𝑏) = 〈1o, 〈𝑎, 𝑏〉〉 |
6 | df-goal 32610 | . . . . 5 ⊢ ∀𝑔𝑖𝑢 = 〈2o, 〈𝑖, 𝑢〉〉 | |
7 | 5, 6 | eqeq12i 2835 | . . . 4 ⊢ ((𝑎⊼𝑔𝑏) = ∀𝑔𝑖𝑢 ↔ 〈1o, 〈𝑎, 𝑏〉〉 = 〈2o, 〈𝑖, 𝑢〉〉) |
8 | 1oex 8103 | . . . . 5 ⊢ 1o ∈ V | |
9 | opex 5349 | . . . . 5 ⊢ 〈𝑎, 𝑏〉 ∈ V | |
10 | 8, 9 | opth 5361 | . . . 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 3061 | . 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 1536 ≠ wne 3015 Vcvv 3491 〈cop 4566 (class class class)co 7149 1oc1o 8088 2oc2o 8089 ⊼𝑔cgna 32602 ∀𝑔cgol 32603 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7152 df-om 7574 df-1o 8095 df-2o 8096 df-gona 32609 df-goal 32610 |
This theorem is referenced by: gonarlem 32662 gonar 32663 goalrlem 32664 goalr 32665 fmlasucdisj 32667 |
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