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Theorem 0elsucexmid 4547
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 4501 . . . 4 {𝑦 ∈ {∅} ∣ 𝜑} ∈ On
2 0elsucexmid.1 . . . 4 𝑥 ∈ On ∅ ∈ suc 𝑥
3 suceq 4385 . . . . . 6 (𝑥 = {𝑦 ∈ {∅} ∣ 𝜑} → suc 𝑥 = suc {𝑦 ∈ {∅} ∣ 𝜑})
43eleq2d 2240 . . . . 5 (𝑥 = {𝑦 ∈ {∅} ∣ 𝜑} → (∅ ∈ suc 𝑥 ↔ ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑}))
54rspcv 2830 . . . 4 ({𝑦 ∈ {∅} ∣ 𝜑} ∈ On → (∀𝑥 ∈ On ∅ ∈ suc 𝑥 → ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑}))
61, 2, 5mp2 16 . . 3 ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑}
7 0ex 4114 . . . 4 ∅ ∈ V
87elsuc 4389 . . 3 (∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑} ↔ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑}))
96, 8mpbi 144 . 2 (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑})
107snid 3612 . . . . 5 ∅ ∈ {∅}
11 biidd 171 . . . . . 6 (𝑦 = ∅ → (𝜑𝜑))
1211elrab3 2887 . . . . 5 (∅ ∈ {∅} → (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
1310, 12ax-mp 5 . . . 4 (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
1413biimpi 119 . . 3 (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} → 𝜑)
15 ordtriexmidlem2 4502 . . . 4 ({𝑦 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
1615eqcoms 2173 . . 3 (∅ = {𝑦 ∈ {∅} ∣ 𝜑} → ¬ 𝜑)
1714, 16orim12i 754 . 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 703   = wceq 1348  wcel 2141  wral 2448  {crab 2452  c0 3414  {csn 3581  Oncon0 4346  suc csuc 4348
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-nul 4113  ax-pow 4158
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-uni 3795  df-tr 4086  df-iord 4349  df-on 4351  df-suc 4354
This theorem is referenced by: (None)
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