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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 4364 | . . . 4 ⊢ {𝑦 ∈ {∅} ∣ 𝜑} ∈ On | |
2 | 0elsucexmid.1 | . . . 4 ⊢ ∀𝑥 ∈ On ∅ ∈ suc 𝑥 | |
3 | suceq 4253 | . . . . . 6 ⊢ (𝑥 = {𝑦 ∈ {∅} ∣ 𝜑} → suc 𝑥 = suc {𝑦 ∈ {∅} ∣ 𝜑}) | |
4 | 3 | eleq2d 2164 | . . . . 5 ⊢ (𝑥 = {𝑦 ∈ {∅} ∣ 𝜑} → (∅ ∈ suc 𝑥 ↔ ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑})) |
5 | 4 | rspcv 2732 | . . . 4 ⊢ ({𝑦 ∈ {∅} ∣ 𝜑} ∈ On → (∀𝑥 ∈ On ∅ ∈ suc 𝑥 → ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑})) |
6 | 1, 2, 5 | mp2 16 | . . 3 ⊢ ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑} |
7 | 0ex 3987 | . . . 4 ⊢ ∅ ∈ V | |
8 | 7 | elsuc 4257 | . . 3 ⊢ (∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑} ↔ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑})) |
9 | 6, 8 | mpbi 144 | . 2 ⊢ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑}) |
10 | 7 | snid 3495 | . . . . 5 ⊢ ∅ ∈ {∅} |
11 | biidd 171 | . . . . . 6 ⊢ (𝑦 = ∅ → (𝜑 ↔ 𝜑)) | |
12 | 11 | elrab3 2786 | . . . . 5 ⊢ (∅ ∈ {∅} → (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ↔ 𝜑)) |
13 | 10, 12 | ax-mp 7 | . . . 4 ⊢ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ↔ 𝜑) |
14 | 13 | biimpi 119 | . . 3 ⊢ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} → 𝜑) |
15 | ordtriexmidlem2 4365 | . . . 4 ⊢ ({𝑦 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑) | |
16 | 15 | eqcoms 2098 | . . 3 ⊢ (∅ = {𝑦 ∈ {∅} ∣ 𝜑} → ¬ 𝜑) |
17 | 14, 16 | orim12i 714 | . 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 104 ∨ wo 667 = wceq 1296 ∈ wcel 1445 ∀wral 2370 {crab 2374 ∅c0 3302 {csn 3466 Oncon0 4214 suc csuc 4216 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-nul 3986 ax-pow 4030 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-rab 2379 df-v 2635 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-uni 3676 df-tr 3959 df-iord 4217 df-on 4219 df-suc 4222 |
This theorem is referenced by: (None) |
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