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Theorem 0elsucexmid 4347
Description: If the successor of any ordinal class contains the empty set, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2021.)
Hypothesis
Ref Expression
0elsucexmid.1 𝑥 ∈ On ∅ ∈ suc 𝑥
Assertion
Ref Expression
0elsucexmid (𝜑 ∨ ¬ 𝜑)
Distinct variable group:   𝜑,𝑥

Proof of Theorem 0elsucexmid
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ordtriexmidlem 4302 . . . 4 {𝑦 ∈ {∅} ∣ 𝜑} ∈ On
2 0elsucexmid.1 . . . 4 𝑥 ∈ On ∅ ∈ suc 𝑥
3 suceq 4196 . . . . . 6 (𝑥 = {𝑦 ∈ {∅} ∣ 𝜑} → suc 𝑥 = suc {𝑦 ∈ {∅} ∣ 𝜑})
43eleq2d 2154 . . . . 5 (𝑥 = {𝑦 ∈ {∅} ∣ 𝜑} → (∅ ∈ suc 𝑥 ↔ ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑}))
54rspcv 2710 . . . 4 ({𝑦 ∈ {∅} ∣ 𝜑} ∈ On → (∀𝑥 ∈ On ∅ ∈ suc 𝑥 → ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑}))
61, 2, 5mp2 16 . . 3 ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑}
7 0ex 3934 . . . 4 ∅ ∈ V
87elsuc 4200 . . 3 (∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑} ↔ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑}))
96, 8mpbi 143 . 2 (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑})
107snid 3452 . . . . 5 ∅ ∈ {∅}
11 biidd 170 . . . . . 6 (𝑦 = ∅ → (𝜑𝜑))
1211elrab3 2762 . . . . 5 (∅ ∈ {∅} → (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
1310, 12ax-mp 7 . . . 4 (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
1413biimpi 118 . . 3 (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} → 𝜑)
15 ordtriexmidlem2 4303 . . . 4 ({𝑦 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
1615eqcoms 2088 . . 3 (∅ = {𝑦 ∈ {∅} ∣ 𝜑} → ¬ 𝜑)
1714, 16orim12i 709 . 2 ((∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑}) → (𝜑 ∨ ¬ 𝜑))
189, 17ax-mp 7 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 103  wo 662   = wceq 1287  wcel 1436  wral 2355  {crab 2359  c0 3272  {csn 3425  Oncon0 4157  suc csuc 4159
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-nul 3933  ax-pow 3977
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2616  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-pw 3411  df-sn 3431  df-uni 3631  df-tr 3905  df-iord 4160  df-on 4162  df-suc 4165
This theorem is referenced by: (None)
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