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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 35395 | . . . 4 ⊢ (𝑥 ∈ ω → ∅ ∉ (Fmla‘𝑥)) | |
| 2 | df-nel 3047 | . . . 4 ⊢ (∅ ∉ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘𝑥)) | |
| 3 | 1, 2 | sylib 218 | . . 3 ⊢ (𝑥 ∈ ω → ¬ ∅ ∈ (Fmla‘𝑥)) |
| 4 | 3 | nrex 3074 | . 2 ⊢ ¬ ∃𝑥 ∈ ω ∅ ∈ (Fmla‘𝑥) |
| 5 | df-nel 3047 | . . 3 ⊢ (∅ ∉ (Fmla‘ω) ↔ ¬ ∅ ∈ (Fmla‘ω)) | |
| 6 | fmla 35386 | . . . . 5 ⊢ (Fmla‘ω) = ∪ 𝑥 ∈ ω (Fmla‘𝑥) | |
| 7 | 6 | eleq2i 2833 | . . . 4 ⊢ (∅ ∈ (Fmla‘ω) ↔ ∅ ∈ ∪ 𝑥 ∈ ω (Fmla‘𝑥)) |
| 8 | eliun 4995 | . . . 4 ⊢ (∅ ∈ ∪ 𝑥 ∈ ω (Fmla‘𝑥) ↔ ∃𝑥 ∈ ω ∅ ∈ (Fmla‘𝑥)) | |
| 9 | 7, 8 | bitri 275 | . . 3 ⊢ (∅ ∈ (Fmla‘ω) ↔ ∃𝑥 ∈ ω ∅ ∈ (Fmla‘𝑥)) |
| 10 | 5, 9 | xchbinx 334 | . 2 ⊢ (∅ ∉ (Fmla‘ω) ↔ ¬ ∃𝑥 ∈ ω ∅ ∈ (Fmla‘𝑥)) |
| 11 | 4, 10 | mpbir 231 | 1 ⊢ ∅ ∉ (Fmla‘ω) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∈ wcel 2108 ∉ wnel 3046 ∃wrex 3070 ∅c0 4333 ∪ ciun 4991 ‘cfv 6561 ωcom 7887 Fmlacfmla 35342 |
| 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 |
| 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-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-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-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-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-map 8868 df-goel 35345 df-gona 35346 df-goal 35347 df-sat 35348 df-fmla 35350 |
| This theorem is referenced by: (None) |
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