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Theorem undifexmid 3991
Description: Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3344 and undifdcss 6558 are provable, the full statement implies excluded middle as shown here. (Contributed by Jim Kingdon, 16-Jun-2022.)
Hypothesis
Ref Expression
undifexmid.1 (𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) = 𝑦)
Assertion
Ref Expression
undifexmid (𝜑 ∨ ¬ 𝜑)
Distinct variable group:   𝜑,𝑥,𝑦

Proof of Theorem undifexmid
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 0ex 3931 . . . . 5 ∅ ∈ V
21snid 3449 . . . 4 ∅ ∈ {∅}
3 ssrab2 3090 . . . . 5 {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}
4 p0ex 3986 . . . . . . 7 {∅} ∈ V
54rabex 3948 . . . . . 6 {𝑧 ∈ {∅} ∣ 𝜑} ∈ V
6 sseq12 3033 . . . . . . 7 ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → (𝑥𝑦 ↔ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}))
7 simpl 107 . . . . . . . . 9 ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → 𝑥 = {𝑧 ∈ {∅} ∣ 𝜑})
8 simpr 108 . . . . . . . . . 10 ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → 𝑦 = {∅})
98, 7difeq12d 3103 . . . . . . . . 9 ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → (𝑦𝑥) = ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}))
107, 9uneq12d 3139 . . . . . . . 8 ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → (𝑥 ∪ (𝑦𝑥)) = ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})))
1110, 8eqeq12d 2097 . . . . . . 7 ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → ((𝑥 ∪ (𝑦𝑥)) = 𝑦 ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})) = {∅}))
126, 11bibi12d 233 . . . . . 6 ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → ((𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) = 𝑦) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})) = {∅})))
13 undifexmid.1 . . . . . 6 (𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) = 𝑦)
145, 4, 12, 13vtocl2 2665 . . . . 5 ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})) = {∅})
153, 14mpbi 143 . . . 4 ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})) = {∅}
162, 15eleqtrri 2158 . . 3 ∅ ∈ ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}))
17 elun 3125 . . 3 (∅ ∈ ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})) ↔ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})))
1816, 17mpbi 143 . 2 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}))
19 biidd 170 . . . . . 6 (𝑧 = ∅ → (𝜑𝜑))
2019elrab3 2760 . . . . 5 (∅ ∈ {∅} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
212, 20ax-mp 7 . . . 4 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
2221biimpi 118 . . 3 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
23 eldifn 3107 . . . 4 (∅ ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) → ¬ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})
2423, 21sylnib 634 . . 3 (∅ ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) → ¬ 𝜑)
2522, 24orim12i 709 . 2 ((∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})) → (𝜑 ∨ ¬ 𝜑))
2618, 25ax-mp 7 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 102  wb 103  wo 662   = wceq 1285  wcel 1434  {crab 2357  cdif 2981  cun 2982  wss 2984  c0 3269  {csn 3422
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-nul 3930  ax-pow 3974
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rab 2362  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428
This theorem is referenced by: (None)
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