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Mirrors > Home > ILE Home > Th. List > Mathboxes > sbthomlem | GIF version |
Description: Lemma for sbthom 13221. (Contributed by Mario Carneiro and Jim Kingdon, 13-Jul-2023.) |
Ref | Expression |
---|---|
sbthomlem.lpo | ⊢ (𝜑 → ω ∈ Omni) |
sbthomlem.y | ⊢ (𝜑 → 𝑌 ⊆ {∅}) |
sbthomlem.f | ⊢ (𝜑 → 𝐹:ω–1-1-onto→(𝑌 ⊔ ω)) |
Ref | Expression |
---|---|
sbthomlem | ⊢ (𝜑 → (𝑌 = ∅ ∨ 𝑌 = {∅})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbthomlem.lpo | . . . 4 ⊢ (𝜑 → ω ∈ Omni) | |
2 | sbthomlem.f | . . . . 5 ⊢ (𝜑 → 𝐹:ω–1-1-onto→(𝑌 ⊔ ω)) | |
3 | f1ofo 5374 | . . . . 5 ⊢ (𝐹:ω–1-1-onto→(𝑌 ⊔ ω) → 𝐹:ω–onto→(𝑌 ⊔ ω)) | |
4 | 2, 3 | syl 14 | . . . 4 ⊢ (𝜑 → 𝐹:ω–onto→(𝑌 ⊔ ω)) |
5 | 1, 4 | fodjuomni 7021 | . . 3 ⊢ (𝜑 → (∃𝑧 𝑧 ∈ 𝑌 ∨ 𝑌 = ∅)) |
6 | 5 | orcomd 718 | . 2 ⊢ (𝜑 → (𝑌 = ∅ ∨ ∃𝑧 𝑧 ∈ 𝑌)) |
7 | sbthomlem.y | . . . 4 ⊢ (𝜑 → 𝑌 ⊆ {∅}) | |
8 | sssnm 3681 | . . . 4 ⊢ (∃𝑧 𝑧 ∈ 𝑌 → (𝑌 ⊆ {∅} ↔ 𝑌 = {∅})) | |
9 | 7, 8 | syl5ibcom 154 | . . 3 ⊢ (𝜑 → (∃𝑧 𝑧 ∈ 𝑌 → 𝑌 = {∅})) |
10 | 9 | orim2d 777 | . 2 ⊢ (𝜑 → ((𝑌 = ∅ ∨ ∃𝑧 𝑧 ∈ 𝑌) → (𝑌 = ∅ ∨ 𝑌 = {∅}))) |
11 | 6, 10 | mpd 13 | 1 ⊢ (𝜑 → (𝑌 = ∅ ∨ 𝑌 = {∅})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 697 = wceq 1331 ∃wex 1468 ∈ wcel 1480 ⊆ wss 3071 ∅c0 3363 {csn 3527 ωcom 4504 –onto→wfo 5121 –1-1-onto→wf1o 5122 ⊔ cdju 6922 Omnicomni 7004 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-1o 6313 df-2o 6314 df-map 6544 df-dju 6923 df-inl 6932 df-inr 6933 df-omni 7006 |
This theorem is referenced by: sbthom 13221 |
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