Theorem 1sdom2dom 9090

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 8789	. . . 4 ⊢ Rel ≺
2	1	brrelex2i 5662	. . 3 ⊢ (1o ≺ 𝐴 → 𝐴 ∈ V)
3		sdomdom 8819	. . . . . . 7 ⊢ (1o ≺ 𝐴 → 1o ≼ 𝐴)
4		0sdom1dom 9081	. . . . . . 7 ⊢ (∅ ≺ 𝐴 ↔ 1o ≼ 𝐴)
5	3, 4	sylibr 233	. . . . . 6 ⊢ (1o ≺ 𝐴 → ∅ ≺ 𝐴)
6		0sdomg 8947	. . . . . . 7 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
7	2, 6	syl 17	. . . . . 6 ⊢ (1o ≺ 𝐴 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
8	5, 7	mpbid 231	. . . . 5 ⊢ (1o ≺ 𝐴 → 𝐴 ≠ ∅)
9		n0snor2el 4775	. . . . 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 8820	. . . . 5 ⊢ (1o ≺ 𝐴 → ¬ 1o ≈ 𝐴)
12		df1o2 8352	. . . . . . . 8 ⊢ 1o = {∅}
13		0ex 5245	. . . . . . . . 9 ⊢ ∅ ∈ V
14		vex 3444	. . . . . . . . 9 ⊢ 𝑥 ∈ V
15		en2sn 8884	. . . . . . . . 9 ⊢ ((∅ ∈ V ∧ 𝑥 ∈ V) → {∅} ≈ {𝑥})
16	13, 14, 15	mp2an 689	. . . . . . . 8 ⊢ {∅} ≈ {𝑥}
17	12, 16	eqbrtri 5107	. . . . . . 7 ⊢ 1o ≈ {𝑥}
18		breq2 5090	. . . . . . 7 ⊢ (𝐴 = {𝑥} → (1o ≈ 𝐴 ↔ 1o ≈ {𝑥}))
19	17, 18	mpbiri 257	. . . . . 6 ⊢ (𝐴 = {𝑥} → 1o ≈ 𝐴)
20	19	exlimiv 1932	. . . . 5 ⊢ (∃𝑥 𝐴 = {𝑥} → 1o ≈ 𝐴)
21	11, 20	nsyl 140	. . . 4 ⊢ (1o ≺ 𝐴 → ¬ ∃𝑥 𝐴 = {𝑥})
22	10, 21	olcnd 874	. . 3 ⊢ (1o ≺ 𝐴 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦)
23		rex2dom 9089	. . 3 ⊢ ((𝐴 ∈ V ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦) → 2o ≼ 𝐴)
24	2, 22, 23	syl2anc 584	. 2 ⊢ (1o ≺ 𝐴 → 2o ≼ 𝐴)
25		snsspr1 4758	. . . . 5 ⊢ {∅} ⊆ {∅, 1o}
26		df2o3 8353	. . . . 5 ⊢ 2o = {∅, 1o}
27	25, 12, 26	3sstr4i 3973	. . . 4 ⊢ 1o ⊆ 2o
28		domssl 8837	. . . 4 ⊢ ((1o ⊆ 2o ∧ 2o ≼ 𝐴) → 1o ≼ 𝐴)
29	27, 28	mpan 687	. . 3 ⊢ (2o ≼ 𝐴 → 1o ≼ 𝐴)
30		snnen2o 9080	. . . . . . . . . . . 12 ⊢ ¬ {𝑦} ≈ 2o
31	13	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → ∅ ∈ V)
32		1oex 8355	. . . . . . . . . . . . . . . . 17 ⊢ 1o ∈ V
33	32	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → 1o ∈ V)
34		1n0 8367	. . . . . . . . . . . . . . . . . 18 ⊢ 1o ≠ ∅
35	34	nesymi 2998	. . . . . . . . . . . . . . . . 17 ⊢ ¬ ∅ = 1o
36	35	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → ¬ ∅ = 1o)
37	31, 33, 36	enpr2d 8892	. . . . . . . . . . . . . . 15 ⊢ (⊤ → {∅, 1o} ≈ 2o)
38	37	mptru 1547	. . . . . . . . . . . . . 14 ⊢ {∅, 1o} ≈ 2o
39	26, 38	eqbrtri 5107	. . . . . . . . . . . . 13 ⊢ 2o ≈ 2o
40		breq1 5089	. . . . . . . . . . . . 13 ⊢ (2o = {𝑦} → (2o ≈ 2o ↔ {𝑦} ≈ 2o))
41	39, 40	mpbii 232	. . . . . . . . . . . 12 ⊢ (2o = {𝑦} → {𝑦} ≈ 2o)
42	30, 41	mto 196	. . . . . . . . . . 11 ⊢ ¬ 2o = {𝑦}
43	42	nex 1801	. . . . . . . . . 10 ⊢ ¬ ∃𝑦2o = {𝑦}
44		2on0 8361	. . . . . . . . . . 11 ⊢ 2o ≠ ∅
45		f1cdmsn 7193	. . . . . . . . . . 11 ⊢ ((𝑓:2o–1-1→{𝑥} ∧ 2o ≠ ∅) → ∃𝑦2o = {𝑦})
46	44, 45	mpan2 688	. . . . . . . . . 10 ⊢ (𝑓:2o–1-1→{𝑥} → ∃𝑦2o = {𝑦})
47	43, 46	mto 196	. . . . . . . . 9 ⊢ ¬ 𝑓:2o–1-1→{𝑥}
48	47	nex 1801	. . . . . . . 8 ⊢ ¬ ∃𝑓 𝑓:2o–1-1→{𝑥}
49		brdomi 8797	. . . . . . . 8 ⊢ (2o ≼ {𝑥} → ∃𝑓 𝑓:2o–1-1→{𝑥})
50	48, 49	mto 196	. . . . . . 7 ⊢ ¬ 2o ≼ {𝑥}
51		breq2 5090	. . . . . . 7 ⊢ (𝐴 = {𝑥} → (2o ≼ 𝐴 ↔ 2o ≼ {𝑥}))
52	50, 51	mtbiri 326	. . . . . 6 ⊢ (𝐴 = {𝑥} → ¬ 2o ≼ 𝐴)
53	52	con2i 139	. . . . 5 ⊢ (2o ≼ 𝐴 → ¬ 𝐴 = {𝑥})
54	53	nexdv 1938	. . . 4 ⊢ (2o ≼ 𝐴 → ¬ ∃𝑥 𝐴 = {𝑥})
55		reldom 8788	. . . . . . 7 ⊢ Rel ≼
56	55	brrelex2i 5662	. . . . . 6 ⊢ (2o ≼ 𝐴 → 𝐴 ∈ V)
57		breng 8791	. . . . . . 7 ⊢ ((1o ∈ V ∧ 𝐴 ∈ V) → (1o ≈ 𝐴 ↔ ∃𝑓 𝑓:1o–1-1-onto→𝐴))
58	32, 57	mpan 687	. . . . . 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 233	. . . . . . 7 ⊢ (2o ≼ 𝐴 → ∅ ≺ 𝐴)
61	56, 6	syl 17	. . . . . . 7 ⊢ (2o ≼ 𝐴 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
62	60, 61	mpbid 231	. . . . . 6 ⊢ (2o ≼ 𝐴 → 𝐴 ≠ ∅)
63		f1ocnv 6765	. . . . . . . . . 10 ⊢ (𝑓:1o–1-1-onto→𝐴 → ◡𝑓:𝐴–1-1-onto→1o)
64		f1of1 6752	. . . . . . . . . . 11 ⊢ (◡𝑓:𝐴–1-1-onto→1o → ◡𝑓:𝐴–1-1→1o)
65		f1eq3 6704	. . . . . . . . . . . 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 217	. . . . . . . . . 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 7193	. . . . . . . . 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 414	. . . . . . 7 ⊢ (𝐴 ≠ ∅ → (𝑓:1o–1-1-onto→𝐴 → ∃𝑥 𝐴 = {𝑥}))
72	71	exlimdv 1935	. . . . . 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 239	. . . 4 ⊢ (2o ≼ 𝐴 → (1o ≈ 𝐴 → ∃𝑥 𝐴 = {𝑥}))
75	54, 74	mtod 197	. . 3 ⊢ (2o ≼ 𝐴 → ¬ 1o ≈ 𝐴)
76		brsdom 8814	. . 3 ⊢ (1o ≺ 𝐴 ↔ (1o ≼ 𝐴 ∧ ¬ 1o ≈ 𝐴))
77	29, 75, 76	sylanbrc 583	. 2 ⊢ (2o ≼ 𝐴 → 1o ≺ 𝐴)
78	24, 77	impbii 208	1 ⊢ (1o ≺ 𝐴 ↔ 2o ≼ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∨ wo 844   = wceq 1540  ⊤wtru 1541  ∃wex 1780   ∈ wcel 2105   ≠ wne 2940  ∃wrex 3070  Vcvv 3440   ⊆ wss 3896  ∅c0 4266  {csn 4570  {cpr 4572   class class class wbr 5086  ^◡ccnv 5606  –1-1→wf1 6462  –1-1-onto→wf1o 6464  1_oc1o 8338  2_oc2o 8339   ≈ cen 8779   ≼ cdom 8780   ≺ csdm 8781

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5237  ax-nul 5244  ax-pr 5366

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3442  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4850  df-br 5087  df-opab 5149  df-id 5506  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-suc 6294  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473  df-1o 8345  df-2o 8346  df-en 8783  df-dom 8784  df-sdom 8785

This theorem is referenced by:  1sdom  9091