Theorem 1sdom2dom 9283

Description: Strict dominance over 1 is the same as dominance over 2. (Contributed by BTernaryTau, 23-Dec-2024.)

Assertion

Ref	Expression
1sdom2dom	⊢ (1o ≺ 𝐴 ↔ 2o ≼ 𝐴)

Proof of Theorem 1sdom2dom

Dummy variables 𝑥 𝑦 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relsdom 9018	. . . 4 ⊢ Rel ≺
2	1	brrelex2i 5715	. . 3 ⊢ (1o ≺ 𝐴 → 𝐴 ∈ V)
3		sdomdom 9045	. . . . . . 7 ⊢ (1o ≺ 𝐴 → 1o ≼ 𝐴)
4		0sdom1dom 9275	. . . . . . 7 ⊢ (∅ ≺ 𝐴 ↔ 1o ≼ 𝐴)
5	3, 4	sylibr 237	. . . . . 6 ⊢ (1o ≺ 𝐴 → ∅ ≺ 𝐴)
6		0sdomg 9163	. . . . . . 7 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
7	2, 6	syl 18	. . . . . 6 ⊢ (1o ≺ 𝐴 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
8	5, 7	mpbid 235	. . . . 5 ⊢ (1o ≺ 𝐴 → 𝐴 ≠ ∅)
9		n0snor2el 4793	. . . . 5 ⊢ (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦 ∨ ∃𝑥 𝐴 = {𝑥}))
10	8, 9	syl 18	. . . 4 ⊢ (1o ≺ 𝐴 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦 ∨ ∃𝑥 𝐴 = {𝑥}))
11		sdomnen 9046	. . . . 5 ⊢ (1o ≺ 𝐴 → ¬ 1o ≈ 𝐴)
12		df1o2 8521	. . . . . . . 8 ⊢ 1o = {∅}
13		0ex 5267	. . . . . . . . 9 ⊢ ∅ ∈ V
14		vex 3458	. . . . . . . . 9 ⊢ 𝑥 ∈ V
15		en2sn 9107	. . . . . . . . 9 ⊢ ((∅ ∈ V ∧ 𝑥 ∈ V) → {∅} ≈ {𝑥})
16	13, 14, 15	mp2an 705	. . . . . . . 8 ⊢ {∅} ≈ {𝑥}
17	12, 16	eqbrtri 5132	. . . . . . 7 ⊢ 1o ≈ {𝑥}
18		breq2 5112	. . . . . . 7 ⊢ (𝐴 = {𝑥} → (1o ≈ 𝐴 ↔ 1o ≈ {𝑥}))
19	17, 18	mpbiri 261	. . . . . 6 ⊢ (𝐴 = {𝑥} → 1o ≈ 𝐴)
20	19	exlimiv 1965	. . . . 5 ⊢ (∃𝑥 𝐴 = {𝑥} → 1o ≈ 𝐴)
21	11, 20	nsyl 141	. . . 4 ⊢ (1o ≺ 𝐴 → ¬ ∃𝑥 𝐴 = {𝑥})
22	10, 21	olcnd 891	. . 3 ⊢ (1o ≺ 𝐴 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦)
23		rex2dom 9282	. . 3 ⊢ ((𝐴 ∈ V ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦) → 2o ≼ 𝐴)
24	2, 22, 23	syl2anc 596	. 2 ⊢ (1o ≺ 𝐴 → 2o ≼ 𝐴)
25		snsspr1 4775	. . . . 5 ⊢ {∅} ⊆ {∅, 1o}
26		df2o3 8522	. . . . 5 ⊢ 2o = {∅, 1o}
27	25, 12, 26	3sstr4i 3964	. . . 4 ⊢ 1o ⊆ 2o
28		domssl 9063	. . . 4 ⊢ ((1o ⊆ 2o ∧ 2o ≼ 𝐴) → 1o ≼ 𝐴)
29	27, 28	mpan 703	. . 3 ⊢ (2o ≼ 𝐴 → 1o ≼ 𝐴)
30		snnen2o 9274	. . . . . . . . . . . 12 ⊢ ¬ {𝑦} ≈ 2o
31	13	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → ∅ ∈ V)
32		1oex 8524	. . . . . . . . . . . . . . . . 17 ⊢ 1o ∈ V
33	32	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → 1o ∈ V)
34		1n0 8533	. . . . . . . . . . . . . . . . . 18 ⊢ 1o ≠ ∅
35	34	nesymi 3016	. . . . . . . . . . . . . . . . 17 ⊢ ¬ ∅ = 1o
36	35	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → ¬ ∅ = 1o)
37	31, 33, 36	enpr2d 9114	. . . . . . . . . . . . . . 15 ⊢ (⊤ → {∅, 1o} ≈ 2o)
38	37	mptru 1577	. . . . . . . . . . . . . 14 ⊢ {∅, 1o} ≈ 2o
39	26, 38	eqbrtri 5132	. . . . . . . . . . . . 13 ⊢ 2o ≈ 2o
40		breq1 5111	. . . . . . . . . . . . 13 ⊢ (2o = {𝑦} → (2o ≈ 2o ↔ {𝑦} ≈ 2o))
41	39, 40	mpbii 236	. . . . . . . . . . . 12 ⊢ (2o = {𝑦} → {𝑦} ≈ 2o)
42	30, 41	mto 200	. . . . . . . . . . 11 ⊢ ¬ 2o = {𝑦}
43	42	nex 1833	. . . . . . . . . 10 ⊢ ¬ ∃𝑦2o = {𝑦}
44		2on0 8529	. . . . . . . . . . 11 ⊢ 2o ≠ ∅
45		f1cdmsn 7321	. . . . . . . . . . 11 ⊢ ((𝑓:2o⟶1-1{𝑥} ∧ 2o ≠ ∅) → ∃𝑦2o = {𝑦})
46	44, 45	mpan2 704	. . . . . . . . . 10 ⊢ (𝑓:2o⟶1-1{𝑥} → ∃𝑦2o = {𝑦})
47	43, 46	mto 200	. . . . . . . . 9 ⊢ ¬ 𝑓:2o⟶1-1{𝑥}
48	47	nex 1833	. . . . . . . 8 ⊢ ¬ ∃𝑓 𝑓:2o⟶1-1{𝑥}
49		brdomi 9024	. . . . . . . 8 ⊢ (2o ≼ {𝑥} → ∃𝑓 𝑓:2o⟶1-1{𝑥})
50	48, 49	mto 200	. . . . . . 7 ⊢ ¬ 2o ≼ {𝑥}
51		breq2 5112	. . . . . . 7 ⊢ (𝐴 = {𝑥} → (2o ≼ 𝐴 ↔ 2o ≼ {𝑥}))
52	50, 51	mtbiri 330	. . . . . 6 ⊢ (𝐴 = {𝑥} → ¬ 2o ≼ 𝐴)
53	52	con2i 140	. . . . 5 ⊢ (2o ≼ 𝐴 → ¬ 𝐴 = {𝑥})
54	53	nexdv 1971	. . . 4 ⊢ (2o ≼ 𝐴 → ¬ ∃𝑥 𝐴 = {𝑥})
55		reldom 9017	. . . . . . 7 ⊢ Rel ≼
56	55	brrelex2i 5715	. . . . . 6 ⊢ (2o ≼ 𝐴 → 𝐴 ∈ V)
57		breng 9020	. . . . . . 7 ⊢ ((1o ∈ V ∧ 𝐴 ∈ V) → (1o ≈ 𝐴 ↔ ∃𝑓 𝑓:1o⟶1-1onto𝐴))
58	32, 57	mpan 703	. . . . . 6 ⊢ (𝐴 ∈ V → (1o ≈ 𝐴 ↔ ∃𝑓 𝑓:1o⟶1-1onto𝐴))
59	56, 58	syl 18	. . . . 5 ⊢ (2o ≼ 𝐴 → (1o ≈ 𝐴 ↔ ∃𝑓 𝑓:1o⟶1-1onto𝐴))
60	29, 4	sylibr 237	. . . . . . 7 ⊢ (2o ≼ 𝐴 → ∅ ≺ 𝐴)
61	56, 6	syl 18	. . . . . . 7 ⊢ (2o ≼ 𝐴 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
62	60, 61	mpbid 235	. . . . . 6 ⊢ (2o ≼ 𝐴 → 𝐴 ≠ ∅)
63		f1ocnv 6867	. . . . . . . . . 10 ⊢ (𝑓:1o⟶1-1onto𝐴 → ◡𝑓:𝐴⟶1-1onto1o)
64		f1of1 6853	. . . . . . . . . . 11 ⊢ (◡𝑓:𝐴⟶1-1onto1o → ◡𝑓:𝐴⟶1-11o)
65		f1eq3 6805	. . . . . . . . . . . 12 ⊢ (1o = {∅} → (◡𝑓:𝐴⟶1-11o ↔ ◡𝑓:𝐴⟶1-1{∅}))
66	12, 65	ax-mp 5	. . . . . . . . . . 11 ⊢ (◡𝑓:𝐴⟶1-11o ↔ ◡𝑓:𝐴⟶1-1{∅})
67	64, 66	sylib 221	. . . . . . . . . 10 ⊢ (◡𝑓:𝐴⟶1-1onto1o → ◡𝑓:𝐴⟶1-1{∅})
68	63, 67	syl 18	. . . . . . . . 9 ⊢ (𝑓:1o⟶1-1onto𝐴 → ◡𝑓:𝐴⟶1-1{∅})
69		f1cdmsn 7321	. . . . . . . . 9 ⊢ ((◡𝑓:𝐴⟶1-1{∅} ∧ 𝐴 ≠ ∅) → ∃𝑥 𝐴 = {𝑥})
70	68, 69	sylan 592	. . . . . . . 8 ⊢ ((𝑓:1o⟶1-1onto𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 𝐴 = {𝑥})
71	70	expcom 419	. . . . . . 7 ⊢ (𝐴 ≠ ∅ → (𝑓:1o⟶1-1onto𝐴 → ∃𝑥 𝐴 = {𝑥}))
72	71	exlimdv 1968	. . . . . 6 ⊢ (𝐴 ≠ ∅ → (∃𝑓 𝑓:1o⟶1-1onto𝐴 → ∃𝑥 𝐴 = {𝑥}))
73	62, 72	syl 18	. . . . 5 ⊢ (2o ≼ 𝐴 → (∃𝑓 𝑓:1o⟶1-1onto𝐴 → ∃𝑥 𝐴 = {𝑥}))
74	59, 73	sylbid 243	. . . 4 ⊢ (2o ≼ 𝐴 → (1o ≈ 𝐴 → ∃𝑥 𝐴 = {𝑥}))
75	54, 74	mtod 201	. . 3 ⊢ (2o ≼ 𝐴 → ¬ 1o ≈ 𝐴)
76		brsdom 9039	. . 3 ⊢ (1o ≺ 𝐴 ↔ (1o ≼ 𝐴 ∧ ¬ 1o ≈ 𝐴))
77	29, 75, 76	sylanbrc 595	. 2 ⊢ (2o ≼ 𝐴 → 1o ≺ 𝐴)
78	24, 77	impbii 212	1 ⊢ (1o ≺ 𝐴 ↔ 2o ≼ 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∨ wo 861   = wceq 1570  ⊤wtru 1571  ∃wex 1812   ∈ wcel 2147   ≠ wne 2959  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  {csn 4588  {cpr 4590   class class class wbr 5108  ^◡ccnv 5657  ⟶1-1wf1 6563  ⟶1-1ontowf1o 6565  1_oc1o 8507  2_oc2o 8508   ≈ cen 9008   ≼ cdom 9009   ≺ csdm 9010

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-1o 8514  df-2o 8515  df-en 9012  df-dom 9013  df-sdom 9014

This theorem is used by:  1sdom  9284