Theorem ordtri2or2exmidlem 4558

Description: A set which is 2_o 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.

Step	Hyp	Ref	Expression
1		simpll 527	. . . . . . 7 ⊢ (((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = ∅) → 𝑦 ∈ 𝑧)
2		noel 3450	. . . . . . . . 9 ⊢ ¬ 𝑦 ∈ ∅
3		eleq2 2257	. . . . . . . . 9 ⊢ (𝑧 = ∅ → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ ∅))
4	2, 3	mtbiri 676	. . . . . . . 8 ⊢ (𝑧 = ∅ → ¬ 𝑦 ∈ 𝑧)
5	4	adantl 277	. . . . . . 7 ⊢ (((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = ∅) → ¬ 𝑦 ∈ 𝑧)
6	1, 5	pm2.21dd 621	. . . . . 6 ⊢ (((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = ∅) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
7		eleq2 2257	. . . . . . . . . . 11 ⊢ (𝑧 = {∅} → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ {∅}))
8	7	biimpac 298	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 = {∅}) → 𝑦 ∈ {∅})
9		velsn 3635	. . . . . . . . . 10 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
10	8, 9	sylib 122	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 = {∅}) → 𝑦 = ∅)
11		orc 713	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → (𝑦 = ∅ ∨ 𝑦 = {∅}))
12		vex 2763	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
13	12	elpr 3639	. . . . . . . . . 10 ⊢ (𝑦 ∈ {∅, {∅}} ↔ (𝑦 = ∅ ∨ 𝑦 = {∅}))
14	11, 13	sylibr 134	. . . . . . . . 9 ⊢ (𝑦 = ∅ → 𝑦 ∈ {∅, {∅}})
15	10, 14	syl 14	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 = {∅}) → 𝑦 ∈ {∅, {∅}})
16	15	adantlr 477	. . . . . . 7 ⊢ (((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = {∅}) → 𝑦 ∈ {∅, {∅}})
17		biidd 172	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜑))
18	17	elrab 2916	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ↔ (𝑧 ∈ {∅, {∅}} ∧ 𝜑))
19	18	simprbi 275	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} → 𝜑)
20	19	ad2antlr 489	. . . . . . 7 ⊢ (((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = {∅}) → 𝜑)
21		biidd 172	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜑))
22	21	elrab 2916	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ↔ (𝑦 ∈ {∅, {∅}} ∧ 𝜑))
23	16, 20, 22	sylanbrc 417	. . . . . 6 ⊢ (((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = {∅}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
24		elrabi 2913	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} → 𝑧 ∈ {∅, {∅}})
25		vex 2763	. . . . . . . . 9 ⊢ 𝑧 ∈ V
26	25	elpr 3639	. . . . . . . 8 ⊢ (𝑧 ∈ {∅, {∅}} ↔ (𝑧 = ∅ ∨ 𝑧 = {∅}))
27	24, 26	sylib 122	. . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} → (𝑧 = ∅ ∨ 𝑧 = {∅}))
28	27	adantl 277	. . . . . 6 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) → (𝑧 = ∅ ∨ 𝑧 = {∅}))
29	6, 23, 28	mpjaodan 799	. . . . 5 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
30	29	gen2 1461	. . . 4 ⊢ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
31		dftr2 4129	. . . 4 ⊢ (Tr {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}))
32	30, 31	mpbir 146	. . 3 ⊢ Tr {𝑥 ∈ {∅, {∅}} ∣ 𝜑}
33		ssrab2 3264	. . 3 ⊢ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅, {∅}}
34		2ordpr 4556	. . 3 ⊢ Ord {∅, {∅}}
35		trssord 4411	. . 3 ⊢ ((Tr {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∧ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅, {∅}} ∧ Ord {∅, {∅}}) → Ord {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
36	32, 33, 34, 35	mp3an 1348	. 2 ⊢ Ord {𝑥 ∈ {∅, {∅}} ∣ 𝜑}
37		pp0ex 4218	. . . 4 ⊢ {∅, {∅}} ∈ V
38	37	rabex 4173	. . 3 ⊢ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ V
39	38	elon 4405	. 2 ⊢ ({𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ On ↔ Ord {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
40	36, 39	mpbir 146	1 ⊢ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ On

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 709  ∀wal 1362   = wceq 1364   ∈ wcel 2164  {crab 2476   ⊆ wss 3153  ∅c0 3446  {csn 3618  {cpr 3619  Tr wtr 4127  Ord word 4393  Oncon0 4394

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-uni 3836  df-tr 4128  df-iord 4397  df-on 4399  df-suc 4402

This theorem is referenced by:  ordtri2or2exmid  4603  ontri2orexmidim  4604