Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ennnfonelemdc | GIF version |
Description: Lemma for ennnfone 12165. A direct consequence of fidcenumlemrk 6898. (Contributed by Jim Kingdon, 15-Jul-2023.) |
Ref | Expression |
---|---|
ennnfonelemdc.dceq | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦) |
ennnfonelemdc.f | ⊢ (𝜑 → 𝐹:ω–onto→𝐴) |
ennnfonelemdc.p | ⊢ (𝜑 → 𝑃 ∈ ω) |
Ref | Expression |
---|---|
ennnfonelemdc | ⊢ (𝜑 → DECID (𝐹‘𝑃) ∈ (𝐹 “ 𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ennnfonelemdc.dceq | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦) | |
2 | ennnfonelemdc.f | . . 3 ⊢ (𝜑 → 𝐹:ω–onto→𝐴) | |
3 | ennnfonelemdc.p | . . 3 ⊢ (𝜑 → 𝑃 ∈ ω) | |
4 | omelon 4568 | . . . . 5 ⊢ ω ∈ On | |
5 | 4 | onelssi 4389 | . . . 4 ⊢ (𝑃 ∈ ω → 𝑃 ⊆ ω) |
6 | 3, 5 | syl 14 | . . 3 ⊢ (𝜑 → 𝑃 ⊆ ω) |
7 | fof 5392 | . . . . 5 ⊢ (𝐹:ω–onto→𝐴 → 𝐹:ω⟶𝐴) | |
8 | 2, 7 | syl 14 | . . . 4 ⊢ (𝜑 → 𝐹:ω⟶𝐴) |
9 | 8, 3 | ffvelrnd 5603 | . . 3 ⊢ (𝜑 → (𝐹‘𝑃) ∈ 𝐴) |
10 | 1, 2, 3, 6, 9 | fidcenumlemrk 6898 | . 2 ⊢ (𝜑 → ((𝐹‘𝑃) ∈ (𝐹 “ 𝑃) ∨ ¬ (𝐹‘𝑃) ∈ (𝐹 “ 𝑃))) |
11 | df-dc 821 | . 2 ⊢ (DECID (𝐹‘𝑃) ∈ (𝐹 “ 𝑃) ↔ ((𝐹‘𝑃) ∈ (𝐹 “ 𝑃) ∨ ¬ (𝐹‘𝑃) ∈ (𝐹 “ 𝑃))) | |
12 | 10, 11 | sylibr 133 | 1 ⊢ (𝜑 → DECID (𝐹‘𝑃) ∈ (𝐹 “ 𝑃)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 698 DECID wdc 820 ∈ wcel 2128 ∀wral 2435 ⊆ wss 3102 ωcom 4549 “ cima 4589 ⟶wf 5166 –onto→wfo 5168 ‘cfv 5170 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-iinf 4547 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-opab 4026 df-tr 4063 df-id 4253 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-fo 5176 df-fv 5178 |
This theorem is referenced by: ennnfonelemg 12143 ennnfonelemp1 12146 ennnfonelemss 12150 ennnfonelemkh 12152 ennnfonelemhf1o 12153 |
Copyright terms: Public domain | W3C validator |