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Theorem ordtri2or2exmidlem 4278
 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 489 . . . . . . 7 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = ∅) → 𝑦𝑧)
2 noel 3255 . . . . . . . . 9 ¬ 𝑦 ∈ ∅
3 eleq2 2117 . . . . . . . . 9 (𝑧 = ∅ → (𝑦𝑧𝑦 ∈ ∅))
42, 3mtbiri 610 . . . . . . . 8 (𝑧 = ∅ → ¬ 𝑦𝑧)
54adantl 266 . . . . . . 7 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = ∅) → ¬ 𝑦𝑧)
61, 5pm2.21dd 560 . . . . . 6 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = ∅) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
7 eleq2 2117 . . . . . . . . . . 11 (𝑧 = {∅} → (𝑦𝑧𝑦 ∈ {∅}))
87biimpac 286 . . . . . . . . . 10 ((𝑦𝑧𝑧 = {∅}) → 𝑦 ∈ {∅})
9 velsn 3419 . . . . . . . . . 10 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
108, 9sylib 131 . . . . . . . . 9 ((𝑦𝑧𝑧 = {∅}) → 𝑦 = ∅)
11 orc 643 . . . . . . . . . 10 (𝑦 = ∅ → (𝑦 = ∅ ∨ 𝑦 = {∅}))
12 vex 2577 . . . . . . . . . . 11 𝑦 ∈ V
1312elpr 3423 . . . . . . . . . 10 (𝑦 ∈ {∅, {∅}} ↔ (𝑦 = ∅ ∨ 𝑦 = {∅}))
1411, 13sylibr 141 . . . . . . . . 9 (𝑦 = ∅ → 𝑦 ∈ {∅, {∅}})
1510, 14syl 14 . . . . . . . 8 ((𝑦𝑧𝑧 = {∅}) → 𝑦 ∈ {∅, {∅}})
1615adantlr 454 . . . . . . 7 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = {∅}) → 𝑦 ∈ {∅, {∅}})
17 biidd 165 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝜑𝜑))
1817elrab 2720 . . . . . . . . 9 (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ↔ (𝑧 ∈ {∅, {∅}} ∧ 𝜑))
1918simprbi 264 . . . . . . . 8 (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} → 𝜑)
2019ad2antlr 466 . . . . . . 7 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = {∅}) → 𝜑)
21 biidd 165 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜑))
2221elrab 2720 . . . . . . 7 (𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ↔ (𝑦 ∈ {∅, {∅}} ∧ 𝜑))
2316, 20, 22sylanbrc 402 . . . . . 6 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = {∅}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
24 elrabi 2717 . . . . . . . 8 (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} → 𝑧 ∈ {∅, {∅}})
25 vex 2577 . . . . . . . . 9 𝑧 ∈ V
2625elpr 3423 . . . . . . . 8 (𝑧 ∈ {∅, {∅}} ↔ (𝑧 = ∅ ∨ 𝑧 = {∅}))
2724, 26sylib 131 . . . . . . 7 (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} → (𝑧 = ∅ ∨ 𝑧 = {∅}))
2827adantl 266 . . . . . 6 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) → (𝑧 = ∅ ∨ 𝑧 = {∅}))
296, 23, 28mpjaodan 722 . . . . 5 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
3029gen2 1355 . . . 4 𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
31 dftr2 3883 . . . 4 (Tr {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ↔ ∀𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}))
3230, 31mpbir 138 . . 3 Tr {𝑥 ∈ {∅, {∅}} ∣ 𝜑}
33 ssrab2 3052 . . 3 {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅, {∅}}
34 2ordpr 4276 . . 3 Ord {∅, {∅}}
35 trssord 4144 . . 3 ((Tr {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∧ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅, {∅}} ∧ Ord {∅, {∅}}) → Ord {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
3632, 33, 34, 35mp3an 1243 . 2 Ord {𝑥 ∈ {∅, {∅}} ∣ 𝜑}
37 pp0ex 3967 . . . 4 {∅, {∅}} ∈ V
3837rabex 3928 . . 3 {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ V
3938elon 4138 . 2 ({𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ On ↔ Ord {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
4036, 39mpbir 138 1 {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ On
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 101   ∨ wo 639  ∀wal 1257   = wceq 1259   ∈ wcel 1409  {crab 2327   ⊆ wss 2944  ∅c0 3251  {csn 3402  {cpr 3403  Tr wtr 3881  Ord word 4126  Oncon0 4127 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-nul 3910  ax-pow 3954 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-uni 3608  df-tr 3882  df-iord 4130  df-on 4132  df-suc 4135 This theorem is referenced by:  ordtri2or2exmid  4323
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