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Theorem 0elsucexmid 4480
 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 4435 . . . 4 {𝑦 ∈ {∅} ∣ 𝜑} ∈ On
2 0elsucexmid.1 . . . 4 𝑥 ∈ On ∅ ∈ suc 𝑥
3 suceq 4324 . . . . . 6 (𝑥 = {𝑦 ∈ {∅} ∣ 𝜑} → suc 𝑥 = suc {𝑦 ∈ {∅} ∣ 𝜑})
43eleq2d 2209 . . . . 5 (𝑥 = {𝑦 ∈ {∅} ∣ 𝜑} → (∅ ∈ suc 𝑥 ↔ ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑}))
54rspcv 2785 . . . 4 ({𝑦 ∈ {∅} ∣ 𝜑} ∈ On → (∀𝑥 ∈ On ∅ ∈ suc 𝑥 → ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑}))
61, 2, 5mp2 16 . . 3 ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑}
7 0ex 4055 . . . 4 ∅ ∈ V
87elsuc 4328 . . 3 (∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑} ↔ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑}))
96, 8mpbi 144 . 2 (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑})
107snid 3556 . . . . 5 ∅ ∈ {∅}
11 biidd 171 . . . . . 6 (𝑦 = ∅ → (𝜑𝜑))
1211elrab3 2841 . . . . 5 (∅ ∈ {∅} → (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
1310, 12ax-mp 5 . . . 4 (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
1413biimpi 119 . . 3 (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} → 𝜑)
15 ordtriexmidlem2 4436 . . . 4 ({𝑦 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
1615eqcoms 2142 . . 3 (∅ = {𝑦 ∈ {∅} ∣ 𝜑} → ¬ 𝜑)
1714, 16orim12i 748 . 2 ((∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑}) → (𝜑 ∨ ¬ 𝜑))
189, 17ax-mp 5 1 (𝜑 ∨ ¬ 𝜑)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ↔ wb 104   ∨ wo 697   = wceq 1331   ∈ wcel 1480  ∀wral 2416  {crab 2420  ∅c0 3363  {csn 3527  Oncon0 4285  suc csuc 4287 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-uni 3737  df-tr 4027  df-iord 4288  df-on 4290  df-suc 4293 This theorem is referenced by: (None)
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