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Theorem ss1oel2o 15040
Description: Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4210 which more directly illustrates the contrast with el2oss1o 6458. (Contributed by Jim Kingdon, 8-Aug-2022.)
Assertion
Ref Expression
ss1oel2o (EXMID ↔ ∀𝑥(𝑥 ⊆ 1o𝑥 ∈ 2o))

Proof of Theorem ss1oel2o
StepHypRef Expression
1 exmid01 4210 . 2 (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
2 df1o2 6444 . . . . 5 1o = {∅}
32sseq2i 3194 . . . 4 (𝑥 ⊆ 1o𝑥 ⊆ {∅})
4 df2o2 6446 . . . . . 6 2o = {∅, {∅}}
54eleq2i 2254 . . . . 5 (𝑥 ∈ 2o𝑥 ∈ {∅, {∅}})
6 vex 2752 . . . . . 6 𝑥 ∈ V
76elpr 3625 . . . . 5 (𝑥 ∈ {∅, {∅}} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
85, 7bitri 184 . . . 4 (𝑥 ∈ 2o ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
93, 8imbi12i 239 . . 3 ((𝑥 ⊆ 1o𝑥 ∈ 2o) ↔ (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
109albii 1480 . 2 (∀𝑥(𝑥 ⊆ 1o𝑥 ∈ 2o) ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
111, 10bitr4i 187 1 (EXMID ↔ ∀𝑥(𝑥 ⊆ 1o𝑥 ∈ 2o))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wo 709  wal 1361   = wceq 1363  wcel 2158  wss 3141  c0 3434  {csn 3604  {cpr 3605  EXMIDwem 4206  1oc1o 6424  2oc2o 6425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169  ax-nul 4141
This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-sn 3610  df-pr 3611  df-exmid 4207  df-suc 4383  df-1o 6431  df-2o 6432
This theorem is referenced by: (None)
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