|   | Mathbox for Mario Carneiro | < Previous  
      Next > Nearby theorems | |
| 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 7912 | . . . . . 6 ⊢ (𝑍 ∈ ω → suc 𝑍 ∈ ω) | |
| 3 | 1, 2 | ifcld 4572 | . . . . 5 ⊢ (𝑍 ∈ ω → if(𝑥 = 2o, 𝑍, suc 𝑍) ∈ ω) | 
| 4 | 3 | adantr 480 | . . . 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 7134 | . . 3 ⊢ (𝑍 ∈ ω → 𝑆:ω⟶ω) | 
| 7 | omex 9683 | . . . . 5 ⊢ ω ∈ V | |
| 8 | 7 | a1i 11 | . . . 4 ⊢ (𝑍 ∈ ω → ω ∈ V) | 
| 9 | 8, 8 | elmapd 8880 | . . 3 ⊢ (𝑍 ∈ ω → (𝑆 ∈ (ω ↑m ω) ↔ 𝑆:ω⟶ω)) | 
| 10 | 6, 9 | mpbird 257 | . 2 ⊢ (𝑍 ∈ ω → 𝑆 ∈ (ω ↑m ω)) | 
| 11 | sucidg 6465 | . . 3 ⊢ (𝑍 ∈ ω → 𝑍 ∈ suc 𝑍) | |
| 12 | 5 | a1i 11 | . . . 4 ⊢ (𝑍 ∈ ω → 𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 2o, 𝑍, suc 𝑍))) | 
| 13 | iftrue 4531 | . . . . 5 ⊢ (𝑥 = 2o → if(𝑥 = 2o, 𝑍, suc 𝑍) = 𝑍) | |
| 14 | 13 | adantl 481 | . . . 4 ⊢ ((𝑍 ∈ ω ∧ 𝑥 = 2o) → if(𝑥 = 2o, 𝑍, suc 𝑍) = 𝑍) | 
| 15 | 2onn 8680 | . . . . 5 ⊢ 2o ∈ ω | |
| 16 | 15 | a1i 11 | . . . 4 ⊢ (𝑍 ∈ ω → 2o ∈ ω) | 
| 17 | 12, 14, 16, 1 | fvmptd 7023 | . . 3 ⊢ (𝑍 ∈ ω → (𝑆‘2o) = 𝑍) | 
| 18 | 1one2o 8684 | . . . . . . . 8 ⊢ 1o ≠ 2o | |
| 19 | 18 | neii 2942 | . . . . . . 7 ⊢ ¬ 1o = 2o | 
| 20 | eqeq1 2741 | . . . . . . 7 ⊢ (𝑥 = 1o → (𝑥 = 2o ↔ 1o = 2o)) | |
| 21 | 19, 20 | mtbiri 327 | . . . . . 6 ⊢ (𝑥 = 1o → ¬ 𝑥 = 2o) | 
| 22 | 21 | iffalsed 4536 | . . . . 5 ⊢ (𝑥 = 1o → if(𝑥 = 2o, 𝑍, suc 𝑍) = suc 𝑍) | 
| 23 | 22 | adantl 481 | . . . 4 ⊢ ((𝑍 ∈ ω ∧ 𝑥 = 1o) → if(𝑥 = 2o, 𝑍, suc 𝑍) = suc 𝑍) | 
| 24 | 1onn 8678 | . . . . 5 ⊢ 1o ∈ ω | |
| 25 | 24 | a1i 11 | . . . 4 ⊢ (𝑍 ∈ ω → 1o ∈ ω) | 
| 26 | 12, 23, 25, 2 | fvmptd 7023 | . . 3 ⊢ (𝑍 ∈ ω → (𝑆‘1o) = suc 𝑍) | 
| 27 | 11, 17, 26 | 3eltr4d 2856 | . 2 ⊢ (𝑍 ∈ ω → (𝑆‘2o) ∈ (𝑆‘1o)) | 
| 28 | 15, 24 | pm3.2i 470 | . . . 4 ⊢ (2o ∈ ω ∧ 1o ∈ ω) | 
| 29 | 7, 28 | pm3.2i 470 | . . 3 ⊢ (ω ∈ V ∧ (2o ∈ ω ∧ 1o ∈ ω)) | 
| 30 | eqid 2737 | . . . 4 ⊢ (ω Sat∈ (2o∈𝑔1o)) = (ω Sat∈ (2o∈𝑔1o)) | |
| 31 | 30 | sategoelfvb 35424 | . . 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 713 | 1 ⊢ (𝑍 ∈ ω → 𝑆 ∈ (ω Sat∈ (2o∈𝑔1o))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ifcif 4525 ↦ 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) | 
| Copyright terms: Public domain | W3C validator |