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Mirrors > Home > MPE Home > Th. List > Mathboxes > ex-sategoelelomsuc | Structured version Visualization version GIF version |
Description: Example of a valuation of a simplified satisfaction predicate over the ordinal numbers as model for a Godel-set of membership using the properties of a successor: (𝑆‘2o) = 𝑍 ∈ suc 𝑍 = (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-sategoelelomsuc.s | ⊢ 𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 2o, 𝑍, suc 𝑍)) |
Ref | Expression |
---|---|
ex-sategoelelomsuc | ⊢ (𝑍 ∈ ω → 𝑆 ∈ (ω Sat∈ (2o∈𝑔1o))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . . . 6 ⊢ (𝑍 ∈ ω → 𝑍 ∈ ω) | |
2 | peano2 7602 | . . . . . 6 ⊢ (𝑍 ∈ ω → suc 𝑍 ∈ ω) | |
3 | 1, 2 | ifcld 4512 | . . . . 5 ⊢ (𝑍 ∈ ω → if(𝑥 = 2o, 𝑍, suc 𝑍) ∈ ω) |
4 | 3 | adantr 483 | . . . 4 ⊢ ((𝑍 ∈ ω ∧ 𝑥 ∈ ω) → if(𝑥 = 2o, 𝑍, suc 𝑍) ∈ ω) |
5 | ex-sategoelelomsuc.s | . . . 4 ⊢ 𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 2o, 𝑍, suc 𝑍)) | |
6 | 4, 5 | fmptd 6878 | . . 3 ⊢ (𝑍 ∈ ω → 𝑆:ω⟶ω) |
7 | omex 9106 | . . . . 5 ⊢ ω ∈ V | |
8 | 7 | a1i 11 | . . . 4 ⊢ (𝑍 ∈ ω → ω ∈ V) |
9 | 8, 8 | elmapd 8420 | . . 3 ⊢ (𝑍 ∈ ω → (𝑆 ∈ (ω ↑m ω) ↔ 𝑆:ω⟶ω)) |
10 | 6, 9 | mpbird 259 | . 2 ⊢ (𝑍 ∈ ω → 𝑆 ∈ (ω ↑m ω)) |
11 | sucidg 6269 | . . 3 ⊢ (𝑍 ∈ ω → 𝑍 ∈ suc 𝑍) | |
12 | 5 | a1i 11 | . . . 4 ⊢ (𝑍 ∈ ω → 𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 2o, 𝑍, suc 𝑍))) |
13 | iftrue 4473 | . . . . 5 ⊢ (𝑥 = 2o → if(𝑥 = 2o, 𝑍, suc 𝑍) = 𝑍) | |
14 | 13 | adantl 484 | . . . 4 ⊢ ((𝑍 ∈ ω ∧ 𝑥 = 2o) → if(𝑥 = 2o, 𝑍, suc 𝑍) = 𝑍) |
15 | 2onn 8266 | . . . . 5 ⊢ 2o ∈ ω | |
16 | 15 | a1i 11 | . . . 4 ⊢ (𝑍 ∈ ω → 2o ∈ ω) |
17 | 12, 14, 16, 1 | fvmptd 6775 | . . 3 ⊢ (𝑍 ∈ ω → (𝑆‘2o) = 𝑍) |
18 | 1one2o 8269 | . . . . . . . 8 ⊢ 1o ≠ 2o | |
19 | 18 | neii 3018 | . . . . . . 7 ⊢ ¬ 1o = 2o |
20 | eqeq1 2825 | . . . . . . 7 ⊢ (𝑥 = 1o → (𝑥 = 2o ↔ 1o = 2o)) | |
21 | 19, 20 | mtbiri 329 | . . . . . 6 ⊢ (𝑥 = 1o → ¬ 𝑥 = 2o) |
22 | 21 | iffalsed 4478 | . . . . 5 ⊢ (𝑥 = 1o → if(𝑥 = 2o, 𝑍, suc 𝑍) = suc 𝑍) |
23 | 22 | adantl 484 | . . . 4 ⊢ ((𝑍 ∈ ω ∧ 𝑥 = 1o) → if(𝑥 = 2o, 𝑍, suc 𝑍) = suc 𝑍) |
24 | 1onn 8265 | . . . . 5 ⊢ 1o ∈ ω | |
25 | 24 | a1i 11 | . . . 4 ⊢ (𝑍 ∈ ω → 1o ∈ ω) |
26 | 12, 23, 25, 2 | fvmptd 6775 | . . 3 ⊢ (𝑍 ∈ ω → (𝑆‘1o) = suc 𝑍) |
27 | 11, 17, 26 | 3eltr4d 2928 | . 2 ⊢ (𝑍 ∈ ω → (𝑆‘2o) ∈ (𝑆‘1o)) |
28 | 15, 24 | pm3.2i 473 | . . . 4 ⊢ (2o ∈ ω ∧ 1o ∈ ω) |
29 | 7, 28 | pm3.2i 473 | . . 3 ⊢ (ω ∈ V ∧ (2o ∈ ω ∧ 1o ∈ ω)) |
30 | eqid 2821 | . . . 4 ⊢ (ω Sat∈ (2o∈𝑔1o)) = (ω Sat∈ (2o∈𝑔1o)) | |
31 | 30 | sategoelfvb 32666 | . . 3 ⊢ ((ω ∈ V ∧ (2o ∈ ω ∧ 1o ∈ ω)) → (𝑆 ∈ (ω Sat∈ (2o∈𝑔1o)) ↔ (𝑆 ∈ (ω ↑m ω) ∧ (𝑆‘2o) ∈ (𝑆‘1o)))) |
32 | 29, 31 | mp1i 13 | . 2 ⊢ (𝑍 ∈ ω → (𝑆 ∈ (ω Sat∈ (2o∈𝑔1o)) ↔ (𝑆 ∈ (ω ↑m ω) ∧ (𝑆‘2o) ∈ (𝑆‘1o)))) |
33 | 10, 27, 32 | mpbir2and 711 | 1 ⊢ (𝑍 ∈ ω → 𝑆 ∈ (ω Sat∈ (2o∈𝑔1o))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ifcif 4467 ↦ cmpt 5146 suc csuc 6193 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ωcom 7580 1oc1o 8095 2oc2o 8096 ↑m cmap 8406 ∈𝑔cgoe 32580 Sat∈ csate 32585 |
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-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-ac2 9885 |
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-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 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-int 4877 df-iun 4921 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-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-card 9368 df-ac 9542 df-goel 32587 df-gona 32588 df-goal 32589 df-sat 32590 df-sate 32591 df-fmla 32592 |
This theorem is referenced by: (None) |
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