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Theorem 0elsucexmid 4409
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 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})
43eleq2d 2164 . . . . 5 (𝑥 = {𝑦 ∈ {∅} ∣ 𝜑} → (∅ ∈ suc 𝑥 ↔ ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑}))
54rspcv 2732 . . . 4 ({𝑦 ∈ {∅} ∣ 𝜑} ∈ On → (∀𝑥 ∈ On ∅ ∈ suc 𝑥 → ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑}))
61, 2, 5mp2 16 . . 3 ∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑}
7 0ex 3987 . . . 4 ∅ ∈ V
87elsuc 4257 . . 3 (∅ ∈ suc {𝑦 ∈ {∅} ∣ 𝜑} ↔ (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑}))
96, 8mpbi 144 . 2 (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑦 ∈ {∅} ∣ 𝜑})
107snid 3495 . . . . 5 ∅ ∈ {∅}
11 biidd 171 . . . . . 6 (𝑦 = ∅ → (𝜑𝜑))
1211elrab3 2786 . . . . 5 (∅ ∈ {∅} → (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
1310, 12ax-mp 7 . . . 4 (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
1413biimpi 119 . . 3 (∅ ∈ {𝑦 ∈ {∅} ∣ 𝜑} → 𝜑)
15 ordtriexmidlem2 4365 . . . 4 ({𝑦 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
1615eqcoms 2098 . . 3 (∅ = {𝑦 ∈ {∅} ∣ 𝜑} → ¬ 𝜑)
1714, 16orim12i 714 . 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 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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