Theorem 1sdom2dom 9138

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 8876	. . . 4 ⊢ Rel ≺
2	1	brrelex2i 5671	. . 3 ⊢ (1o ≺ 𝐴 → 𝐴 ∈ V)
3		sdomdom 8902	. . . . . . 7 ⊢ (1o ≺ 𝐴 → 1o ≼ 𝐴)
4		0sdom1dom 9130	. . . . . . 7 ⊢ (∅ ≺ 𝐴 ↔ 1o ≼ 𝐴)
5	3, 4	sylibr 234	. . . . . 6 ⊢ (1o ≺ 𝐴 → ∅ ≺ 𝐴)
6		0sdomg 9019	. . . . . . 7 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
7	2, 6	syl 17	. . . . . 6 ⊢ (1o ≺ 𝐴 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
8	5, 7	mpbid 232	. . . . 5 ⊢ (1o ≺ 𝐴 → 𝐴 ≠ ∅)
9		n0snor2el 4782	. . . . 5 ⊢ (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦 ∨ ∃𝑥 𝐴 = {𝑥}))
10	8, 9	syl 17	. . . 4 ⊢ (1o ≺ 𝐴 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦 ∨ ∃𝑥 𝐴 = {𝑥}))
11		sdomnen 8903	. . . . 5 ⊢ (1o ≺ 𝐴 → ¬ 1o ≈ 𝐴)
12		df1o2 8392	. . . . . . . 8 ⊢ 1o = {∅}
13		0ex 5243	. . . . . . . . 9 ⊢ ∅ ∈ V
14		vex 3440	. . . . . . . . 9 ⊢ 𝑥 ∈ V
15		en2sn 8963	. . . . . . . . 9 ⊢ ((∅ ∈ V ∧ 𝑥 ∈ V) → {∅} ≈ {𝑥})
16	13, 14, 15	mp2an 692	. . . . . . . 8 ⊢ {∅} ≈ {𝑥}
17	12, 16	eqbrtri 5110	. . . . . . 7 ⊢ 1o ≈ {𝑥}
18		breq2 5093	. . . . . . 7 ⊢ (𝐴 = {𝑥} → (1o ≈ 𝐴 ↔ 1o ≈ {𝑥}))
19	17, 18	mpbiri 258	. . . . . 6 ⊢ (𝐴 = {𝑥} → 1o ≈ 𝐴)
20	19	exlimiv 1931	. . . . 5 ⊢ (∃𝑥 𝐴 = {𝑥} → 1o ≈ 𝐴)
21	11, 20	nsyl 140	. . . 4 ⊢ (1o ≺ 𝐴 → ¬ ∃𝑥 𝐴 = {𝑥})
22	10, 21	olcnd 877	. . 3 ⊢ (1o ≺ 𝐴 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦)
23		rex2dom 9137	. . 3 ⊢ ((𝐴 ∈ V ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦) → 2o ≼ 𝐴)
24	2, 22, 23	syl2anc 584	. 2 ⊢ (1o ≺ 𝐴 → 2o ≼ 𝐴)
25		snsspr1 4763	. . . . 5 ⊢ {∅} ⊆ {∅, 1o}
26		df2o3 8393	. . . . 5 ⊢ 2o = {∅, 1o}
27	25, 12, 26	3sstr4i 3981	. . . 4 ⊢ 1o ⊆ 2o
28		domssl 8920	. . . 4 ⊢ ((1o ⊆ 2o ∧ 2o ≼ 𝐴) → 1o ≼ 𝐴)
29	27, 28	mpan 690	. . 3 ⊢ (2o ≼ 𝐴 → 1o ≼ 𝐴)
30		snnen2o 9129	. . . . . . . . . . . 12 ⊢ ¬ {𝑦} ≈ 2o
31	13	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → ∅ ∈ V)
32		1oex 8395	. . . . . . . . . . . . . . . . 17 ⊢ 1o ∈ V
33	32	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → 1o ∈ V)
34		1n0 8403	. . . . . . . . . . . . . . . . . 18 ⊢ 1o ≠ ∅
35	34	nesymi 2985	. . . . . . . . . . . . . . . . 17 ⊢ ¬ ∅ = 1o
36	35	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → ¬ ∅ = 1o)
37	31, 33, 36	enpr2d 8970	. . . . . . . . . . . . . . 15 ⊢ (⊤ → {∅, 1o} ≈ 2o)
38	37	mptru 1548	. . . . . . . . . . . . . 14 ⊢ {∅, 1o} ≈ 2o
39	26, 38	eqbrtri 5110	. . . . . . . . . . . . 13 ⊢ 2o ≈ 2o
40		breq1 5092	. . . . . . . . . . . . 13 ⊢ (2o = {𝑦} → (2o ≈ 2o ↔ {𝑦} ≈ 2o))
41	39, 40	mpbii 233	. . . . . . . . . . . 12 ⊢ (2o = {𝑦} → {𝑦} ≈ 2o)
42	30, 41	mto 197	. . . . . . . . . . 11 ⊢ ¬ 2o = {𝑦}
43	42	nex 1801	. . . . . . . . . 10 ⊢ ¬ ∃𝑦2o = {𝑦}
44		2on0 8399	. . . . . . . . . . 11 ⊢ 2o ≠ ∅
45		f1cdmsn 7216	. . . . . . . . . . 11 ⊢ ((𝑓:2o–1-1→{𝑥} ∧ 2o ≠ ∅) → ∃𝑦2o = {𝑦})
46	44, 45	mpan2 691	. . . . . . . . . 10 ⊢ (𝑓:2o–1-1→{𝑥} → ∃𝑦2o = {𝑦})
47	43, 46	mto 197	. . . . . . . . 9 ⊢ ¬ 𝑓:2o–1-1→{𝑥}
48	47	nex 1801	. . . . . . . 8 ⊢ ¬ ∃𝑓 𝑓:2o–1-1→{𝑥}
49		brdomi 8882	. . . . . . . 8 ⊢ (2o ≼ {𝑥} → ∃𝑓 𝑓:2o–1-1→{𝑥})
50	48, 49	mto 197	. . . . . . 7 ⊢ ¬ 2o ≼ {𝑥}
51		breq2 5093	. . . . . . 7 ⊢ (𝐴 = {𝑥} → (2o ≼ 𝐴 ↔ 2o ≼ {𝑥}))
52	50, 51	mtbiri 327	. . . . . 6 ⊢ (𝐴 = {𝑥} → ¬ 2o ≼ 𝐴)
53	52	con2i 139	. . . . 5 ⊢ (2o ≼ 𝐴 → ¬ 𝐴 = {𝑥})
54	53	nexdv 1937	. . . 4 ⊢ (2o ≼ 𝐴 → ¬ ∃𝑥 𝐴 = {𝑥})
55		reldom 8875	. . . . . . 7 ⊢ Rel ≼
56	55	brrelex2i 5671	. . . . . 6 ⊢ (2o ≼ 𝐴 → 𝐴 ∈ V)
57		breng 8878	. . . . . . 7 ⊢ ((1o ∈ V ∧ 𝐴 ∈ V) → (1o ≈ 𝐴 ↔ ∃𝑓 𝑓:1o–1-1-onto→𝐴))
58	32, 57	mpan 690	. . . . . 6 ⊢ (𝐴 ∈ V → (1o ≈ 𝐴 ↔ ∃𝑓 𝑓:1o–1-1-onto→𝐴))
59	56, 58	syl 17	. . . . 5 ⊢ (2o ≼ 𝐴 → (1o ≈ 𝐴 ↔ ∃𝑓 𝑓:1o–1-1-onto→𝐴))
60	29, 4	sylibr 234	. . . . . . 7 ⊢ (2o ≼ 𝐴 → ∅ ≺ 𝐴)
61	56, 6	syl 17	. . . . . . 7 ⊢ (2o ≼ 𝐴 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
62	60, 61	mpbid 232	. . . . . 6 ⊢ (2o ≼ 𝐴 → 𝐴 ≠ ∅)
63		f1ocnv 6775	. . . . . . . . . 10 ⊢ (𝑓:1o–1-1-onto→𝐴 → ◡𝑓:𝐴–1-1-onto→1o)
64		f1of1 6762	. . . . . . . . . . 11 ⊢ (◡𝑓:𝐴–1-1-onto→1o → ◡𝑓:𝐴–1-1→1o)
65		f1eq3 6716	. . . . . . . . . . . 12 ⊢ (1o = {∅} → (◡𝑓:𝐴–1-1→1o ↔ ◡𝑓:𝐴–1-1→{∅}))
66	12, 65	ax-mp 5	. . . . . . . . . . 11 ⊢ (◡𝑓:𝐴–1-1→1o ↔ ◡𝑓:𝐴–1-1→{∅})
67	64, 66	sylib 218	. . . . . . . . . 10 ⊢ (◡𝑓:𝐴–1-1-onto→1o → ◡𝑓:𝐴–1-1→{∅})
68	63, 67	syl 17	. . . . . . . . 9 ⊢ (𝑓:1o–1-1-onto→𝐴 → ◡𝑓:𝐴–1-1→{∅})
69		f1cdmsn 7216	. . . . . . . . 9 ⊢ ((◡𝑓:𝐴–1-1→{∅} ∧ 𝐴 ≠ ∅) → ∃𝑥 𝐴 = {𝑥})
70	68, 69	sylan 580	. . . . . . . 8 ⊢ ((𝑓:1o–1-1-onto→𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 𝐴 = {𝑥})
71	70	expcom 413	. . . . . . 7 ⊢ (𝐴 ≠ ∅ → (𝑓:1o–1-1-onto→𝐴 → ∃𝑥 𝐴 = {𝑥}))
72	71	exlimdv 1934	. . . . . 6 ⊢ (𝐴 ≠ ∅ → (∃𝑓 𝑓:1o–1-1-onto→𝐴 → ∃𝑥 𝐴 = {𝑥}))
73	62, 72	syl 17	. . . . 5 ⊢ (2o ≼ 𝐴 → (∃𝑓 𝑓:1o–1-1-onto→𝐴 → ∃𝑥 𝐴 = {𝑥}))
74	59, 73	sylbid 240	. . . 4 ⊢ (2o ≼ 𝐴 → (1o ≈ 𝐴 → ∃𝑥 𝐴 = {𝑥}))
75	54, 74	mtod 198	. . 3 ⊢ (2o ≼ 𝐴 → ¬ 1o ≈ 𝐴)
76		brsdom 8897	. . 3 ⊢ (1o ≺ 𝐴 ↔ (1o ≼ 𝐴 ∧ ¬ 1o ≈ 𝐴))
77	29, 75, 76	sylanbrc 583	. 2 ⊢ (2o ≼ 𝐴 → 1o ≺ 𝐴)
78	24, 77	impbii 209	1 ⊢ (1o ≺ 𝐴 ↔ 2o ≼ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∨ wo 847   = wceq 1541  ⊤wtru 1542  ∃wex 1780   ∈ wcel 2111   ≠ wne 2928  ∃wrex 3056  Vcvv 3436   ⊆ wss 3897  ∅c0 4280  {csn 4573  {cpr 4575   class class class wbr 5089  ^◡ccnv 5613  –1-1→wf1 6478  –1-1-onto→wf1o 6480  1_oc1o 8378  2_oc2o 8379   ≈ cen 8866   ≼ cdom 8867   ≺ csdm 8868

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-1o 8385  df-2o 8386  df-en 8870  df-dom 8871  df-sdom 8872

This theorem is referenced by:  1sdom  9139