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Theorem ordtri2or2exmidlem 4525
Description: A set which is 2o if 𝜑 or if ¬ 𝜑 is an ordinal. (Contributed by Jim Kingdon, 29-Aug-2021.)
Assertion
Ref Expression
ordtri2or2exmidlem {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ On
Distinct variable group:   𝜑,𝑥

Proof of Theorem ordtri2or2exmidlem
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 527 . . . . . . 7 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = ∅) → 𝑦𝑧)
2 noel 3426 . . . . . . . . 9 ¬ 𝑦 ∈ ∅
3 eleq2 2241 . . . . . . . . 9 (𝑧 = ∅ → (𝑦𝑧𝑦 ∈ ∅))
42, 3mtbiri 675 . . . . . . . 8 (𝑧 = ∅ → ¬ 𝑦𝑧)
54adantl 277 . . . . . . 7 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = ∅) → ¬ 𝑦𝑧)
61, 5pm2.21dd 620 . . . . . 6 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = ∅) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
7 eleq2 2241 . . . . . . . . . . 11 (𝑧 = {∅} → (𝑦𝑧𝑦 ∈ {∅}))
87biimpac 298 . . . . . . . . . 10 ((𝑦𝑧𝑧 = {∅}) → 𝑦 ∈ {∅})
9 velsn 3609 . . . . . . . . . 10 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
108, 9sylib 122 . . . . . . . . 9 ((𝑦𝑧𝑧 = {∅}) → 𝑦 = ∅)
11 orc 712 . . . . . . . . . 10 (𝑦 = ∅ → (𝑦 = ∅ ∨ 𝑦 = {∅}))
12 vex 2740 . . . . . . . . . . 11 𝑦 ∈ V
1312elpr 3613 . . . . . . . . . 10 (𝑦 ∈ {∅, {∅}} ↔ (𝑦 = ∅ ∨ 𝑦 = {∅}))
1411, 13sylibr 134 . . . . . . . . 9 (𝑦 = ∅ → 𝑦 ∈ {∅, {∅}})
1510, 14syl 14 . . . . . . . 8 ((𝑦𝑧𝑧 = {∅}) → 𝑦 ∈ {∅, {∅}})
1615adantlr 477 . . . . . . 7 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = {∅}) → 𝑦 ∈ {∅, {∅}})
17 biidd 172 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝜑𝜑))
1817elrab 2893 . . . . . . . . 9 (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ↔ (𝑧 ∈ {∅, {∅}} ∧ 𝜑))
1918simprbi 275 . . . . . . . 8 (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} → 𝜑)
2019ad2antlr 489 . . . . . . 7 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = {∅}) → 𝜑)
21 biidd 172 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜑))
2221elrab 2893 . . . . . . 7 (𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ↔ (𝑦 ∈ {∅, {∅}} ∧ 𝜑))
2316, 20, 22sylanbrc 417 . . . . . 6 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = {∅}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
24 elrabi 2890 . . . . . . . 8 (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} → 𝑧 ∈ {∅, {∅}})
25 vex 2740 . . . . . . . . 9 𝑧 ∈ V
2625elpr 3613 . . . . . . . 8 (𝑧 ∈ {∅, {∅}} ↔ (𝑧 = ∅ ∨ 𝑧 = {∅}))
2724, 26sylib 122 . . . . . . 7 (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} → (𝑧 = ∅ ∨ 𝑧 = {∅}))
2827adantl 277 . . . . . 6 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) → (𝑧 = ∅ ∨ 𝑧 = {∅}))
296, 23, 28mpjaodan 798 . . . . 5 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
3029gen2 1450 . . . 4 𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
31 dftr2 4103 . . . 4 (Tr {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ↔ ∀𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}))
3230, 31mpbir 146 . . 3 Tr {𝑥 ∈ {∅, {∅}} ∣ 𝜑}
33 ssrab2 3240 . . 3 {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅, {∅}}
34 2ordpr 4523 . . 3 Ord {∅, {∅}}
35 trssord 4380 . . 3 ((Tr {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∧ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅, {∅}} ∧ Ord {∅, {∅}}) → Ord {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
3632, 33, 34, 35mp3an 1337 . 2 Ord {𝑥 ∈ {∅, {∅}} ∣ 𝜑}
37 pp0ex 4189 . . . 4 {∅, {∅}} ∈ V
3837rabex 4147 . . 3 {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ V
3938elon 4374 . 2 ({𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ On ↔ Ord {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
4036, 39mpbir 146 1 {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ On
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 708  wal 1351   = wceq 1353  wcel 2148  {crab 2459  wss 3129  c0 3422  {csn 3592  {cpr 3593  Tr wtr 4101  Ord word 4362  Oncon0 4363
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-uni 3810  df-tr 4102  df-iord 4366  df-on 4368  df-suc 4371
This theorem is referenced by:  ordtri2or2exmid  4570  ontri2orexmidim  4571
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