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Mirrors > Home > ILE Home > Th. List > 0elsucexmid | GIF version |
Description: If the successor of any ordinal class contains the empty set, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2021.) |
Ref | Expression |
---|---|
0elsucexmid.1 | ⊢ ∀𝑥 ∈ On ∅ ∈ suc 𝑥 |
Ref | Expression |
---|---|
0elsucexmid | ⊢ (𝜑 ∨ ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtriexmidlem 4271 | . . . 4 ⊢ {𝑦 ∈ {∅} ∣ 𝜑} ∈ On | |
2 | 0elsucexmid.1 | . . . 4 ⊢ ∀𝑥 ∈ On ∅ ∈ suc 𝑥 | |
3 | suceq 4165 | . . . . . 6 ⊢ (𝑥 = {𝑦 ∈ {∅} ∣ 𝜑} → suc 𝑥 = suc {𝑦 ∈ {∅} ∣ 𝜑}) | |
4 | 3 | eleq2d 2149 | . . . . 5 ⊢ (𝑥 = {𝑦 ∈ {∅} ∣ 𝜑} → (∅ ∈ suc 𝑥 ↔ ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑})) |
5 | 4 | rspcv 2698 | . . . 4 ⊢ ({𝑦 ∈ {∅} ∣ 𝜑} ∈ On → (∀𝑥 ∈ On ∅ ∈ suc 𝑥 → ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑})) |
6 | 1, 2, 5 | mp2 16 | . . 3 ⊢ ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑} |
7 | 0ex 3913 | . . . 4 ⊢ ∅ ∈ V | |
8 | 7 | elsuc 4169 | . . 3 ⊢ (∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑} ↔ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑})) |
9 | 6, 8 | mpbi 143 | . 2 ⊢ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑}) |
10 | 7 | snid 3433 | . . . . 5 ⊢ ∅ ∈ {∅} |
11 | biidd 170 | . . . . . 6 ⊢ (𝑦 = ∅ → (𝜑 ↔ 𝜑)) | |
12 | 11 | elrab3 2751 | . . . . 5 ⊢ (∅ ∈ {∅} → (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ↔ 𝜑)) |
13 | 10, 12 | ax-mp 7 | . . . 4 ⊢ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ↔ 𝜑) |
14 | 13 | biimpi 118 | . . 3 ⊢ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} → 𝜑) |
15 | ordtriexmidlem2 4272 | . . . 4 ⊢ ({𝑦 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑) | |
16 | 15 | eqcoms 2085 | . . 3 ⊢ (∅ = {𝑦 ∈ {∅} ∣ 𝜑} → ¬ 𝜑) |
17 | 14, 16 | orim12i 709 | . 2 ⊢ ((∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑}) → (𝜑 ∨ ¬ 𝜑)) |
18 | 9, 17 | ax-mp 7 | 1 ⊢ (𝜑 ∨ ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 103 ∨ wo 662 = wceq 1285 ∈ wcel 1434 ∀wral 2349 {crab 2353 ∅c0 3258 {csn 3406 Oncon0 4126 suc csuc 4128 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-sep 3904 ax-nul 3912 ax-pow 3956 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1687 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ral 2354 df-rex 2355 df-rab 2358 df-v 2604 df-dif 2976 df-un 2978 df-in 2980 df-ss 2987 df-nul 3259 df-pw 3392 df-sn 3412 df-uni 3610 df-tr 3884 df-iord 4129 df-on 4131 df-suc 4134 |
This theorem is referenced by: (None) |
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