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Mirrors > Home > MPE Home > Th. List > Mathboxes > ex-sategoel | Structured version Visualization version GIF version |
Description: Instance of sategoelfv 34939 for the example of a valuation of a simplified satisfaction predicate for a Godel-set of membership. (Contributed by AV, 5-Nov-2023.) |
Ref | Expression |
---|---|
sategoelfvb.s | ⊢ 𝐸 = (𝑀 Sat∈ (𝐴∈𝑔𝐵)) |
ex-sategoelel.s | ⊢ 𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅))) |
Ref | Expression |
---|---|
ex-sategoel | ⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → (𝑆‘𝐴) ∈ (𝑆‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 764 | . 2 ⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → 𝑀 ∈ WUni) | |
2 | 3simpa 1145 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵) → (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) | |
3 | 2 | adantl 481 | . 2 ⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) |
4 | sategoelfvb.s | . . 3 ⊢ 𝐸 = (𝑀 Sat∈ (𝐴∈𝑔𝐵)) | |
5 | ex-sategoelel.s | . . 3 ⊢ 𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅))) | |
6 | 4, 5 | ex-sategoelel 34940 | . 2 ⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → 𝑆 ∈ 𝐸) |
7 | 4 | sategoelfv 34939 | . 2 ⊢ ((𝑀 ∈ WUni ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑆 ∈ 𝐸) → (𝑆‘𝐴) ∈ (𝑆‘𝐵)) |
8 | 1, 3, 6, 7 | syl3anc 1368 | 1 ⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → (𝑆‘𝐴) ∈ (𝑆‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∅c0 4317 ifcif 4523 𝒫 cpw 4597 ↦ cmpt 5224 ‘cfv 6537 (class class class)co 7405 ωcom 7852 WUnicwun 10697 ∈𝑔cgoe 34852 Sat∈ csate 34857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-ac2 10460 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-ac 10113 df-wun 10699 df-goel 34859 df-gona 34860 df-goal 34861 df-sat 34862 df-sate 34863 df-fmla 34864 |
This theorem is referenced by: (None) |
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