Theorem 0sdom1dom 9268

Description: Strict dominance over 0 is the same as dominance over 1. For a shorter proof requiring ax-un 7778, see 0sdom1domALT . (Contributed by NM, 28-Sep-2004.) Avoid ax-un 7778. (Revised by BTernaryTau, 7-Dec-2024.)

Assertion

Ref	Expression
0sdom1dom	⊢ (∅ ≺ 𝐴 ↔ 1o ≼ 𝐴)

Proof of Theorem 0sdom1dom

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		relsdom 9011	. . 3 ⊢ Rel ≺
2	1	brrelex2i 5713	. 2 ⊢ (∅ ≺ 𝐴 → 𝐴 ∈ V)
3		reldom 9010	. . 3 ⊢ Rel ≼
4	3	brrelex2i 5713	. 2 ⊢ (1o ≼ 𝐴 → 𝐴 ∈ V)
5		0sdomg 9156	. . 3 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
6		n0 4303	. . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
7		snssi 4746	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴)
8		df1o2 8514	. . . . . . . . . . 11 ⊢ 1o = {∅}
9		0ex 5265	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
10		vex 3458	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
11		en2sn 9100	. . . . . . . . . . . 12 ⊢ ((∅ ∈ V ∧ 𝑥 ∈ V) → {∅} ≈ {𝑥})
12	9, 10, 11	mp2an 705	. . . . . . . . . . 11 ⊢ {∅} ≈ {𝑥}
13	8, 12	eqbrtri 5130	. . . . . . . . . 10 ⊢ 1o ≈ {𝑥}
14		endom 9037	. . . . . . . . . 10 ⊢ (1o ≈ {𝑥} → 1o ≼ {𝑥})
15	13, 14	ax-mp 5	. . . . . . . . 9 ⊢ 1o ≼ {𝑥}
16		domssr 9057	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ {𝑥} ⊆ 𝐴 ∧ 1o ≼ {𝑥}) → 1o ≼ 𝐴)
17	15, 16	mp3an3 1480	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ {𝑥} ⊆ 𝐴) → 1o ≼ 𝐴)
18	17	ex 418	. . . . . . 7 ⊢ (𝐴 ∈ V → ({𝑥} ⊆ 𝐴 → 1o ≼ 𝐴))
19	7, 18	syl5 35	. . . . . 6 ⊢ (𝐴 ∈ V → (𝑥 ∈ 𝐴 → 1o ≼ 𝐴))
20	19	exlimdv 1968	. . . . 5 ⊢ (𝐴 ∈ V → (∃𝑥 𝑥 ∈ 𝐴 → 1o ≼ 𝐴))
21	6, 20	biimtrid 245	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ≠ ∅ → 1o ≼ 𝐴))
22		1n0 8526	. . . . . . 7 ⊢ 1o ≠ ∅
23		dom0 9155	. . . . . . 7 ⊢ (1o ≼ ∅ ↔ 1o = ∅)
24	22, 23	nemtbir 3055	. . . . . 6 ⊢ ¬ 1o ≼ ∅
25		breq2 5110	. . . . . 6 ⊢ (𝐴 = ∅ → (1o ≼ 𝐴 ↔ 1o ≼ ∅))
26	24, 25	mtbiri 330	. . . . 5 ⊢ (𝐴 = ∅ → ¬ 1o ≼ 𝐴)
27	26	necon2ai 2988	. . . 4 ⊢ (1o ≼ 𝐴 → 𝐴 ≠ ∅)
28	21, 27	impbid1 228	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ≠ ∅ ↔ 1o ≼ 𝐴))
29	5, 28	bitrd 282	. 2 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 1o ≼ 𝐴))
30	2, 4, 29	pm5.21nii 381	1 ⊢ (∅ ≺ 𝐴 ↔ 1o ≼ 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570  ∃wex 1812   ∈ wcel 2147   ≠ wne 2959  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  {csn 4588   class class class wbr 5106  1_oc1o 8500   ≈ cen 9001   ≼ cdom 9002   ≺ csdm 9003

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-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396

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-sb 2102  df-mo 2567  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 4839  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-suc 6389  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-1o 8507  df-en 9005  df-dom 9006  df-sdom 9007

This theorem is used by:  1sdom2  9270  1sdom2dom  9276  djulepw  10301  fin45  10500  gchxpidm  10784  rankcf  10892  snct  33379