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Theorem 0elsucexmid 4259
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 4217 . . . 4 {𝑦 ∈ {∅} ∣ 𝜑} ∈ On
2 0elsucexmid.1 . . . 4 𝑥 ∈ On ∅ ∈ suc 𝑥
3 suceq 4111 . . . . . 6 (𝑥 = {𝑦 ∈ {∅} ∣ 𝜑} → suc 𝑥 = suc {𝑦 ∈ {∅} ∣ 𝜑})
43eleq2d 2107 . . . . 5 (𝑥 = {𝑦 ∈ {∅} ∣ 𝜑} → (∅ ∈ suc 𝑥 ↔ ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑}))
54rspcv 2649 . . . 4 ({𝑦 ∈ {∅} ∣ 𝜑} ∈ On → (∀𝑥 ∈ On ∅ ∈ suc 𝑥 → ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑}))
61, 2, 5mp2 16 . . 3 ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑}
7 0ex 3881 . . . 4 ∅ ∈ V
87elsuc 4115 . . 3 (∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑} ↔ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑}))
96, 8mpbi 133 . 2 (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑})
107snid 3399 . . . . 5 ∅ ∈ {∅}
11 biidd 161 . . . . . 6 (𝑦 = ∅ → (𝜑𝜑))
1211elrab3 2696 . . . . 5 (∅ ∈ {∅} → (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
1310, 12ax-mp 7 . . . 4 (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
1413biimpi 113 . . 3 (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} → 𝜑)
15 ordtriexmidlem2 4218 . . . 4 ({𝑦 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
1615eqcoms 2043 . . 3 (∅ = {𝑦 ∈ {∅} ∣ 𝜑} → ¬ 𝜑)
1714, 16orim12i 676 . 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 98  wo 629   = wceq 1243  wcel 1393  wral 2303  {crab 2307  c0 3221  {csn 3372  Oncon0 4072  suc csuc 4074
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 3872  ax-nul 3880  ax-pow 3924
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 2308  df-rex 2309  df-rab 2312  df-v 2556  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-uni 3578  df-tr 3852  df-iord 4075  df-on 4077  df-suc 4080
This theorem is referenced by: (None)
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