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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 8423 | . . . . . . . 8 ⊢ 1o ∈ V | |
3 | 2 | prid1 4724 | . . . . . . 7 ⊢ 1o ∈ {1o, 2o} |
4 | 2oex 8424 | . . . . . . . 8 ⊢ 2o ∈ V | |
5 | 4 | prid2 4725 | . . . . . . 7 ⊢ 2o ∈ {1o, 2o} |
6 | 3, 5 | ifcli 4534 | . . . . . 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 7061 | . . . 4 ⊢ 𝑆:ω⟶{1o, 2o} |
9 | prex 5390 | . . . . 5 ⊢ {1o, 2o} ∈ V | |
10 | omex 9584 | . . . . 5 ⊢ ω ∈ V | |
11 | 9, 10 | elmap 8812 | . . . 4 ⊢ (𝑆 ∈ ({1o, 2o} ↑m ω) ↔ 𝑆:ω⟶{1o, 2o}) |
12 | 8, 11 | mpbir 230 | . . 3 ⊢ 𝑆 ∈ ({1o, 2o} ↑m ω) |
13 | 2 | sucid 6400 | . . . . 5 ⊢ 1o ∈ suc 1o |
14 | df-2o 8414 | . . . . 5 ⊢ 2o = suc 1o | |
15 | 13, 14 | eleqtrri 2833 | . . . 4 ⊢ 1o ∈ 2o |
16 | 2onn 8589 | . . . . 5 ⊢ 2o ∈ ω | |
17 | 1onn 8587 | . . . . 5 ⊢ 1o ∈ ω | |
18 | iftrue 4493 | . . . . . 6 ⊢ (𝑥 = 2o → if(𝑥 = 2o, 1o, 2o) = 1o) | |
19 | 18, 1 | fvmptg 6947 | . . . . 5 ⊢ ((2o ∈ ω ∧ 1o ∈ ω) → (𝑆‘2o) = 1o) |
20 | 16, 17, 19 | mp2an 691 | . . . 4 ⊢ (𝑆‘2o) = 1o |
21 | 1one2o 8593 | . . . . . . . . 9 ⊢ 1o ≠ 2o | |
22 | 21 | neii 2942 | . . . . . . . 8 ⊢ ¬ 1o = 2o |
23 | eqeq1 2737 | . . . . . . . 8 ⊢ (𝑥 = 1o → (𝑥 = 2o ↔ 1o = 2o)) | |
24 | 22, 23 | mtbiri 327 | . . . . . . 7 ⊢ (𝑥 = 1o → ¬ 𝑥 = 2o) |
25 | 24 | iffalsed 4498 | . . . . . 6 ⊢ (𝑥 = 1o → if(𝑥 = 2o, 1o, 2o) = 2o) |
26 | 25, 1 | fvmptg 6947 | . . . . 5 ⊢ ((1o ∈ ω ∧ 2o ∈ ω) → (𝑆‘1o) = 2o) |
27 | 17, 16, 26 | mp2an 691 | . . . 4 ⊢ (𝑆‘1o) = 2o |
28 | 15, 20, 27 | 3eltr4i 2847 | . . 3 ⊢ (𝑆‘2o) ∈ (𝑆‘1o) |
29 | 12, 28 | pm3.2i 472 | . 2 ⊢ (𝑆 ∈ ({1o, 2o} ↑m ω) ∧ (𝑆‘2o) ∈ (𝑆‘1o)) |
30 | 16, 17 | pm3.2i 472 | . . 3 ⊢ (2o ∈ ω ∧ 1o ∈ ω) |
31 | eqid 2733 | . . . 4 ⊢ ({1o, 2o} Sat∈ (2o∈𝑔1o)) = ({1o, 2o} Sat∈ (2o∈𝑔1o)) | |
32 | 31 | sategoelfvb 34070 | . . 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 691 | . 2 ⊢ (𝑆 ∈ ({1o, 2o} Sat∈ (2o∈𝑔1o)) ↔ (𝑆 ∈ ({1o, 2o} ↑m ω) ∧ (𝑆‘2o) ∈ (𝑆‘1o))) |
34 | 29, 33 | mpbir 230 | 1 ⊢ 𝑆 ∈ ({1o, 2o} Sat∈ (2o∈𝑔1o)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ifcif 4487 {cpr 4589 ↦ cmpt 5189 suc csuc 6320 ⟶wf 6493 ‘cfv 6497 (class class class)co 7358 ωcom 7803 1oc1o 8406 2oc2o 8407 ↑m cmap 8768 ∈𝑔cgoe 33984 Sat∈ csate 33989 |
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 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9582 ax-ac2 10404 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-2o 8414 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-card 9880 df-ac 10057 df-goel 33991 df-gona 33992 df-goal 33993 df-sat 33994 df-sate 33995 df-fmla 33996 |
This theorem is referenced by: (None) |
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