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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 7913 | . . . . . 6 ⊢ (𝑍 ∈ ω → suc 𝑍 ∈ ω) | |
3 | 1, 2 | ifcld 4577 | . . . . 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 9681 | . . . . 5 ⊢ ω ∈ V | |
8 | 7 | a1i 11 | . . . 4 ⊢ (𝑍 ∈ ω → ω ∈ V) |
9 | 8, 8 | elmapd 8879 | . . 3 ⊢ (𝑍 ∈ ω → (𝑆 ∈ (ω ↑m ω) ↔ 𝑆:ω⟶ω)) |
10 | 6, 9 | mpbird 257 | . 2 ⊢ (𝑍 ∈ ω → 𝑆 ∈ (ω ↑m ω)) |
11 | sucidg 6467 | . . 3 ⊢ (𝑍 ∈ ω → 𝑍 ∈ suc 𝑍) | |
12 | 5 | a1i 11 | . . . 4 ⊢ (𝑍 ∈ ω → 𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 2o, 𝑍, suc 𝑍))) |
13 | iftrue 4537 | . . . . 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 8679 | . . . . 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 8683 | . . . . . . . 8 ⊢ 1o ≠ 2o | |
19 | 18 | neii 2940 | . . . . . . 7 ⊢ ¬ 1o = 2o |
20 | eqeq1 2739 | . . . . . . 7 ⊢ (𝑥 = 1o → (𝑥 = 2o ↔ 1o = 2o)) | |
21 | 19, 20 | mtbiri 327 | . . . . . 6 ⊢ (𝑥 = 1o → ¬ 𝑥 = 2o) |
22 | 21 | iffalsed 4542 | . . . . 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 8677 | . . . . 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 2854 | . 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 2735 | . . . 4 ⊢ (ω Sat∈ (2o∈𝑔1o)) = (ω Sat∈ (2o∈𝑔1o)) | |
31 | 30 | sategoelfvb 35404 | . . 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 1537 ∈ wcel 2106 Vcvv 3478 ifcif 4531 ↦ cmpt 5231 suc csuc 6388 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ωcom 7887 1oc1o 8498 2oc2o 8499 ↑m cmap 8865 ∈𝑔cgoe 35318 Sat∈ csate 35323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-ac2 10501 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-ac 10154 df-goel 35325 df-gona 35326 df-goal 35327 df-sat 35328 df-sate 35329 df-fmla 35330 |
This theorem is referenced by: (None) |
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