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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmlan0 | Structured version Visualization version GIF version |
Description: The empty set is not a Godel formula. (Contributed by AV, 19-Nov-2023.) |
Ref | Expression |
---|---|
fmlan0 | ⊢ ∅ ∉ (Fmla‘ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fmlaomn0 33065 | . . . 4 ⊢ (𝑥 ∈ ω → ∅ ∉ (Fmla‘𝑥)) | |
2 | df-nel 3047 | . . . 4 ⊢ (∅ ∉ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘𝑥)) | |
3 | 1, 2 | sylib 221 | . . 3 ⊢ (𝑥 ∈ ω → ¬ ∅ ∈ (Fmla‘𝑥)) |
4 | 3 | nrex 3188 | . 2 ⊢ ¬ ∃𝑥 ∈ ω ∅ ∈ (Fmla‘𝑥) |
5 | df-nel 3047 | . . 3 ⊢ (∅ ∉ (Fmla‘ω) ↔ ¬ ∅ ∈ (Fmla‘ω)) | |
6 | fmla 33056 | . . . . 5 ⊢ (Fmla‘ω) = ∪ 𝑥 ∈ ω (Fmla‘𝑥) | |
7 | 6 | eleq2i 2829 | . . . 4 ⊢ (∅ ∈ (Fmla‘ω) ↔ ∅ ∈ ∪ 𝑥 ∈ ω (Fmla‘𝑥)) |
8 | eliun 4908 | . . . 4 ⊢ (∅ ∈ ∪ 𝑥 ∈ ω (Fmla‘𝑥) ↔ ∃𝑥 ∈ ω ∅ ∈ (Fmla‘𝑥)) | |
9 | 7, 8 | bitri 278 | . . 3 ⊢ (∅ ∈ (Fmla‘ω) ↔ ∃𝑥 ∈ ω ∅ ∈ (Fmla‘𝑥)) |
10 | 5, 9 | xchbinx 337 | . 2 ⊢ (∅ ∉ (Fmla‘ω) ↔ ¬ ∃𝑥 ∈ ω ∅ ∈ (Fmla‘𝑥)) |
11 | 4, 10 | mpbir 234 | 1 ⊢ ∅ ∉ (Fmla‘ω) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2110 ∉ wnel 3046 ∃wrex 3062 ∅c0 4237 ∪ ciun 4904 ‘cfv 6380 ωcom 7644 Fmlacfmla 33012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-map 8510 df-goel 33015 df-gona 33016 df-goal 33017 df-sat 33018 df-fmla 33020 |
This theorem is referenced by: (None) |
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