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Theorem undifexmid 4077
Description: Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3409 and undifdcss 6764 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 4015 . . . . 5 ∅ ∈ V
21snid 3522 . . . 4 ∅ ∈ {∅}
3 ssrab2 3148 . . . . 5 {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}
4 p0ex 4072 . . . . . . 7 {∅} ∈ V
54rabex 4032 . . . . . 6 {𝑧 ∈ {∅} ∣ 𝜑} ∈ V
6 sseq12 3088 . . . . . . 7 ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → (𝑥𝑦 ↔ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}))
7 simpl 108 . . . . . . . . 9 ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → 𝑥 = {𝑧 ∈ {∅} ∣ 𝜑})
8 simpr 109 . . . . . . . . . 10 ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → 𝑦 = {∅})
98, 7difeq12d 3161 . . . . . . . . 9 ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → (𝑦𝑥) = ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}))
107, 9uneq12d 3197 . . . . . . . 8 ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → (𝑥 ∪ (𝑦𝑥)) = ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})))
1110, 8eqeq12d 2129 . . . . . . 7 ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → ((𝑥 ∪ (𝑦𝑥)) = 𝑦 ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})) = {∅}))
126, 11bibi12d 234 . . . . . 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 2712 . . . . 5 ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})) = {∅})
153, 14mpbi 144 . . . 4 ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})) = {∅}
162, 15eleqtrri 2190 . . 3 ∅ ∈ ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}))
17 elun 3183 . . 3 (∅ ∈ ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})) ↔ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})))
1816, 17mpbi 144 . 2 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}))
19 biidd 171 . . . . . 6 (𝑧 = ∅ → (𝜑𝜑))
2019elrab3 2810 . . . . 5 (∅ ∈ {∅} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
212, 20ax-mp 7 . . . 4 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
2221biimpi 119 . . 3 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
23 eldifn 3165 . . . 4 (∅ ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) → ¬ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})
2423, 21sylnib 648 . . 3 (∅ ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) → ¬ 𝜑)
2522, 24orim12i 731 . 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 103  wb 104  wo 680   = wceq 1314  wcel 1463  {crab 2394  cdif 3034  cun 3035  wss 3037  c0 3329  {csn 3493
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-nul 4014  ax-pow 4058
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rab 2399  df-v 2659  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499
This theorem is referenced by: (None)
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