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Theorem exmidsssn 4125
 Description: Excluded middle is equivalent to the biconditionalized version of sssnr 3680 for sets. (Contributed by Jim Kingdon, 5-Mar-2023.)
Assertion
Ref Expression
exmidsssn (EXMID ↔ ∀𝑥𝑦(𝑥 ⊆ {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})))
Distinct variable group:   𝑥,𝑦

Proof of Theorem exmidsssn
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 0ss 3401 . . . . . . 7 ∅ ⊆ {𝑦}
2 sseq1 3120 . . . . . . 7 (𝑥 = ∅ → (𝑥 ⊆ {𝑦} ↔ ∅ ⊆ {𝑦}))
31, 2mpbiri 167 . . . . . 6 (𝑥 = ∅ → 𝑥 ⊆ {𝑦})
43adantl 275 . . . . 5 ((EXMID𝑥 = ∅) → 𝑥 ⊆ {𝑦})
5 simpr 109 . . . . . 6 ((EXMID𝑥 = ∅) → 𝑥 = ∅)
65orcd 722 . . . . 5 ((EXMID𝑥 = ∅) → (𝑥 = ∅ ∨ 𝑥 = {𝑦}))
74, 62thd 174 . . . 4 ((EXMID𝑥 = ∅) → (𝑥 ⊆ {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})))
8 sssnm 3681 . . . . . 6 (∃𝑧 𝑧𝑥 → (𝑥 ⊆ {𝑦} ↔ 𝑥 = {𝑦}))
9 neq0r 3377 . . . . . . 7 (∃𝑧 𝑧𝑥 → ¬ 𝑥 = ∅)
10 biorf 733 . . . . . . 7 𝑥 = ∅ → (𝑥 = {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})))
119, 10syl 14 . . . . . 6 (∃𝑧 𝑧𝑥 → (𝑥 = {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})))
128, 11bitrd 187 . . . . 5 (∃𝑧 𝑧𝑥 → (𝑥 ⊆ {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})))
1312adantl 275 . . . 4 ((EXMID ∧ ∃𝑧 𝑧𝑥) → (𝑥 ⊆ {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})))
14 exmidn0m 4124 . . . . . 6 (EXMID ↔ ∀𝑥(𝑥 = ∅ ∨ ∃𝑧 𝑧𝑥))
1514biimpi 119 . . . . 5 (EXMID → ∀𝑥(𝑥 = ∅ ∨ ∃𝑧 𝑧𝑥))
161519.21bi 1537 . . . 4 (EXMID → (𝑥 = ∅ ∨ ∃𝑧 𝑧𝑥))
177, 13, 16mpjaodan 787 . . 3 (EXMID → (𝑥 ⊆ {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})))
1817alrimivv 1847 . 2 (EXMID → ∀𝑥𝑦(𝑥 ⊆ {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})))
19 0ex 4055 . . . . . 6 ∅ ∈ V
20 sneq 3538 . . . . . . . 8 (𝑦 = ∅ → {𝑦} = {∅})
2120sseq2d 3127 . . . . . . 7 (𝑦 = ∅ → (𝑥 ⊆ {𝑦} ↔ 𝑥 ⊆ {∅}))
2220eqeq2d 2151 . . . . . . . 8 (𝑦 = ∅ → (𝑥 = {𝑦} ↔ 𝑥 = {∅}))
2322orbi2d 779 . . . . . . 7 (𝑦 = ∅ → ((𝑥 = ∅ ∨ 𝑥 = {𝑦}) ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})))
2421, 23bibi12d 234 . . . . . 6 (𝑦 = ∅ → ((𝑥 ⊆ {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})) ↔ (𝑥 ⊆ {∅} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))))
2519, 24spcv 2779 . . . . 5 (∀𝑦(𝑥 ⊆ {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})) → (𝑥 ⊆ {∅} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})))
2625biimpd 143 . . . 4 (∀𝑦(𝑥 ⊆ {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})) → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
2726alimi 1431 . . 3 (∀𝑥𝑦(𝑥 ⊆ {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})) → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
28 exmid01 4121 . . 3 (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
2927, 28sylibr 133 . 2 (∀𝑥𝑦(𝑥 ⊆ {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})) → EXMID)
3018, 29impbii 125 1 (EXMID ↔ ∀𝑥𝑦(𝑥 ⊆ {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697  ∀wal 1329   = wceq 1331  ∃wex 1468   ⊆ wss 3071  ∅c0 3363  {csn 3527  EXMIDwem 4118 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-dc 820  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-rab 2425  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-exmid 4119 This theorem is referenced by:  exmidsssnc  4126
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