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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ex-sategoelel12 | Structured version Visualization version GIF version | ||
| Description: Example of a valuation of a simplified satisfaction predicate over a proper pair (of ordinal numbers) as model for a Godel-set of membership using the properties of a successor: (𝑆‘2o) = 1o ∈ 2o = (𝑆‘2o). Remark: the indices 1o and 2o are intentionally reversed to distinguish them from elements of the model: (2o∈𝑔1o) should not be confused with 2o ∈ 1o, which is false. (Contributed by AV, 19-Nov-2023.) |
| Ref | Expression |
|---|---|
| ex-sategoelel12.s | ⊢ 𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 2o, 1o, 2o)) |
| Ref | Expression |
|---|---|
| ex-sategoelel12 | ⊢ 𝑆 ∈ ({1o, 2o} Sat∈ (2o∈𝑔1o)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ex-sategoelel12.s | . . . . 5 ⊢ 𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 2o, 1o, 2o)) | |
| 2 | 1oex 8516 | . . . . . . . 8 ⊢ 1o ∈ V | |
| 3 | 2 | prid1 4762 | . . . . . . 7 ⊢ 1o ∈ {1o, 2o} |
| 4 | 2oex 8517 | . . . . . . . 8 ⊢ 2o ∈ V | |
| 5 | 4 | prid2 4763 | . . . . . . 7 ⊢ 2o ∈ {1o, 2o} |
| 6 | 3, 5 | ifcli 4573 | . . . . . 6 ⊢ if(𝑥 = 2o, 1o, 2o) ∈ {1o, 2o} |
| 7 | 6 | a1i 11 | . . . . 5 ⊢ (𝑥 ∈ ω → if(𝑥 = 2o, 1o, 2o) ∈ {1o, 2o}) |
| 8 | 1, 7 | fmpti 7132 | . . . 4 ⊢ 𝑆:ω⟶{1o, 2o} |
| 9 | prex 5437 | . . . . 5 ⊢ {1o, 2o} ∈ V | |
| 10 | omex 9683 | . . . . 5 ⊢ ω ∈ V | |
| 11 | 9, 10 | elmap 8911 | . . . 4 ⊢ (𝑆 ∈ ({1o, 2o} ↑m ω) ↔ 𝑆:ω⟶{1o, 2o}) |
| 12 | 8, 11 | mpbir 231 | . . 3 ⊢ 𝑆 ∈ ({1o, 2o} ↑m ω) |
| 13 | 2 | sucid 6466 | . . . . 5 ⊢ 1o ∈ suc 1o |
| 14 | df-2o 8507 | . . . . 5 ⊢ 2o = suc 1o | |
| 15 | 13, 14 | eleqtrri 2840 | . . . 4 ⊢ 1o ∈ 2o |
| 16 | 2onn 8680 | . . . . 5 ⊢ 2o ∈ ω | |
| 17 | 1onn 8678 | . . . . 5 ⊢ 1o ∈ ω | |
| 18 | iftrue 4531 | . . . . . 6 ⊢ (𝑥 = 2o → if(𝑥 = 2o, 1o, 2o) = 1o) | |
| 19 | 18, 1 | fvmptg 7014 | . . . . 5 ⊢ ((2o ∈ ω ∧ 1o ∈ ω) → (𝑆‘2o) = 1o) |
| 20 | 16, 17, 19 | mp2an 692 | . . . 4 ⊢ (𝑆‘2o) = 1o |
| 21 | 1one2o 8684 | . . . . . . . . 9 ⊢ 1o ≠ 2o | |
| 22 | 21 | neii 2942 | . . . . . . . 8 ⊢ ¬ 1o = 2o |
| 23 | eqeq1 2741 | . . . . . . . 8 ⊢ (𝑥 = 1o → (𝑥 = 2o ↔ 1o = 2o)) | |
| 24 | 22, 23 | mtbiri 327 | . . . . . . 7 ⊢ (𝑥 = 1o → ¬ 𝑥 = 2o) |
| 25 | 24 | iffalsed 4536 | . . . . . 6 ⊢ (𝑥 = 1o → if(𝑥 = 2o, 1o, 2o) = 2o) |
| 26 | 25, 1 | fvmptg 7014 | . . . . 5 ⊢ ((1o ∈ ω ∧ 2o ∈ ω) → (𝑆‘1o) = 2o) |
| 27 | 17, 16, 26 | mp2an 692 | . . . 4 ⊢ (𝑆‘1o) = 2o |
| 28 | 15, 20, 27 | 3eltr4i 2854 | . . 3 ⊢ (𝑆‘2o) ∈ (𝑆‘1o) |
| 29 | 12, 28 | pm3.2i 470 | . 2 ⊢ (𝑆 ∈ ({1o, 2o} ↑m ω) ∧ (𝑆‘2o) ∈ (𝑆‘1o)) |
| 30 | 16, 17 | pm3.2i 470 | . . 3 ⊢ (2o ∈ ω ∧ 1o ∈ ω) |
| 31 | eqid 2737 | . . . 4 ⊢ ({1o, 2o} Sat∈ (2o∈𝑔1o)) = ({1o, 2o} Sat∈ (2o∈𝑔1o)) | |
| 32 | 31 | sategoelfvb 35424 | . . 3 ⊢ (({1o, 2o} ∈ V ∧ (2o ∈ ω ∧ 1o ∈ ω)) → (𝑆 ∈ ({1o, 2o} Sat∈ (2o∈𝑔1o)) ↔ (𝑆 ∈ ({1o, 2o} ↑m ω) ∧ (𝑆‘2o) ∈ (𝑆‘1o)))) |
| 33 | 9, 30, 32 | mp2an 692 | . 2 ⊢ (𝑆 ∈ ({1o, 2o} Sat∈ (2o∈𝑔1o)) ↔ (𝑆 ∈ ({1o, 2o} ↑m ω) ∧ (𝑆‘2o) ∈ (𝑆‘1o))) |
| 34 | 29, 33 | mpbir 231 | 1 ⊢ 𝑆 ∈ ({1o, 2o} Sat∈ (2o∈𝑔1o)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ifcif 4525 {cpr 4628 ↦ cmpt 5225 suc csuc 6386 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ωcom 7887 1oc1o 8499 2oc2o 8500 ↑m cmap 8866 ∈𝑔cgoe 35338 Sat∈ csate 35343 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-ac2 10503 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-ac 10156 df-goel 35345 df-gona 35346 df-goal 35347 df-sat 35348 df-sate 35349 df-fmla 35350 |
| This theorem is referenced by: (None) |
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