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Theorem ss1oel2o 13116
Description: Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4091 which more directly illustrates the contrast with el2oss1o 13115. (Contributed by Jim Kingdon, 8-Aug-2022.)
Assertion
Ref Expression
ss1oel2o (EXMID ↔ ∀𝑥(𝑥 ⊆ 1o𝑥 ∈ 2o))

Proof of Theorem ss1oel2o
StepHypRef Expression
1 exmid01 4091 . 2 (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
2 df1o2 6294 . . . . 5 1o = {∅}
32sseq2i 3094 . . . 4 (𝑥 ⊆ 1o𝑥 ⊆ {∅})
4 df2o2 6296 . . . . . 6 2o = {∅, {∅}}
54eleq2i 2184 . . . . 5 (𝑥 ∈ 2o𝑥 ∈ {∅, {∅}})
6 vex 2663 . . . . . 6 𝑥 ∈ V
76elpr 3518 . . . . 5 (𝑥 ∈ {∅, {∅}} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
85, 7bitri 183 . . . 4 (𝑥 ∈ 2o ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
93, 8imbi12i 238 . . 3 ((𝑥 ⊆ 1o𝑥 ∈ 2o) ↔ (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
109albii 1431 . 2 (∀𝑥(𝑥 ⊆ 1o𝑥 ∈ 2o) ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
111, 10bitr4i 186 1 (EXMID ↔ ∀𝑥(𝑥 ⊆ 1o𝑥 ∈ 2o))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wo 682  wal 1314   = wceq 1316  wcel 1465  wss 3041  c0 3333  {csn 3497  {cpr 3498  EXMIDwem 4088  1oc1o 6274  2oc2o 6275
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-nul 4024
This theorem depends on definitions:  df-bi 116  df-dc 805  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-sn 3503  df-pr 3504  df-exmid 4089  df-suc 4263  df-1o 6281  df-2o 6282
This theorem is referenced by: (None)
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