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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 4245 | . . . 4 ⊢ {𝑦 ∈ {∅} ∣ 𝜑} ∈ On | |
2 | 0elsucexmid.1 | . . . 4 ⊢ ∀𝑥 ∈ On ∅ ∈ suc 𝑥 | |
3 | suceq 4139 | . . . . . 6 ⊢ (𝑥 = {𝑦 ∈ {∅} ∣ 𝜑} → suc 𝑥 = suc {𝑦 ∈ {∅} ∣ 𝜑}) | |
4 | 3 | eleq2d 2107 | . . . . 5 ⊢ (𝑥 = {𝑦 ∈ {∅} ∣ 𝜑} → (∅ ∈ suc 𝑥 ↔ ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑})) |
5 | 4 | rspcv 2652 | . . . 4 ⊢ ({𝑦 ∈ {∅} ∣ 𝜑} ∈ On → (∀𝑥 ∈ On ∅ ∈ suc 𝑥 → ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑})) |
6 | 1, 2, 5 | mp2 16 | . . 3 ⊢ ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑} |
7 | 0ex 3884 | . . . 4 ⊢ ∅ ∈ V | |
8 | 7 | elsuc 4143 | . . 3 ⊢ (∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑} ↔ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑})) |
9 | 6, 8 | mpbi 133 | . 2 ⊢ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑}) |
10 | 7 | snid 3402 | . . . . 5 ⊢ ∅ ∈ {∅} |
11 | biidd 161 | . . . . . 6 ⊢ (𝑦 = ∅ → (𝜑 ↔ 𝜑)) | |
12 | 11 | elrab3 2699 | . . . . 5 ⊢ (∅ ∈ {∅} → (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ↔ 𝜑)) |
13 | 10, 12 | ax-mp 7 | . . . 4 ⊢ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ↔ 𝜑) |
14 | 13 | biimpi 113 | . . 3 ⊢ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} → 𝜑) |
15 | ordtriexmidlem2 4246 | . . . 4 ⊢ ({𝑦 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑) | |
16 | 15 | eqcoms 2043 | . . 3 ⊢ (∅ = {𝑦 ∈ {∅} ∣ 𝜑} → ¬ 𝜑) |
17 | 14, 16 | orim12i 676 | . 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 98 ∨ wo 629 = wceq 1243 ∈ wcel 1393 ∀wral 2306 {crab 2310 ∅c0 3224 {csn 3375 Oncon0 4100 suc csuc 4102 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-uni 3581 df-tr 3855 df-iord 4103 df-on 4105 df-suc 4108 |
This theorem is referenced by: (None) |
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