Mathbox for Mario Carneiro |
< Previous
Next >
Nearby theorems |
||
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 32637 | . . . 4 ⊢ (𝑥 ∈ ω → ∅ ∉ (Fmla‘𝑥)) | |
2 | df-nel 3124 | . . . 4 ⊢ (∅ ∉ (Fmla‘𝑥) ↔ ¬ ∅ ∈ (Fmla‘𝑥)) | |
3 | 1, 2 | sylib 220 | . . 3 ⊢ (𝑥 ∈ ω → ¬ ∅ ∈ (Fmla‘𝑥)) |
4 | 3 | nrex 3269 | . 2 ⊢ ¬ ∃𝑥 ∈ ω ∅ ∈ (Fmla‘𝑥) |
5 | df-nel 3124 | . . 3 ⊢ (∅ ∉ (Fmla‘ω) ↔ ¬ ∅ ∈ (Fmla‘ω)) | |
6 | fmla 32628 | . . . . 5 ⊢ (Fmla‘ω) = ∪ 𝑥 ∈ ω (Fmla‘𝑥) | |
7 | 6 | eleq2i 2904 | . . . 4 ⊢ (∅ ∈ (Fmla‘ω) ↔ ∅ ∈ ∪ 𝑥 ∈ ω (Fmla‘𝑥)) |
8 | eliun 4923 | . . . 4 ⊢ (∅ ∈ ∪ 𝑥 ∈ ω (Fmla‘𝑥) ↔ ∃𝑥 ∈ ω ∅ ∈ (Fmla‘𝑥)) | |
9 | 7, 8 | bitri 277 | . . 3 ⊢ (∅ ∈ (Fmla‘ω) ↔ ∃𝑥 ∈ ω ∅ ∈ (Fmla‘𝑥)) |
10 | 5, 9 | xchbinx 336 | . 2 ⊢ (∅ ∉ (Fmla‘ω) ↔ ¬ ∃𝑥 ∈ ω ∅ ∈ (Fmla‘𝑥)) |
11 | 4, 10 | mpbir 233 | 1 ⊢ ∅ ∉ (Fmla‘ω) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2114 ∉ wnel 3123 ∃wrex 3139 ∅c0 4291 ∪ ciun 4919 ‘cfv 6355 ωcom 7580 Fmlacfmla 32584 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-map 8408 df-goel 32587 df-gona 32588 df-goal 32589 df-sat 32590 df-fmla 32592 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |