Theorem 0sdom1dom 9183

Description: Strict dominance over 0 is the same as dominance over 1. For a shorter proof requiring ax-un 7712, see 0sdom1domALT . (Contributed by NM, 28-Sep-2004.) Avoid ax-un 7712. (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 8927	. . 3 ⊢ Rel ≺
2	1	brrelex2i 5700	. 2 ⊢ (∅ ≺ 𝐴 → 𝐴 ∈ V)
3		reldom 8926	. . 3 ⊢ Rel ≼
4	3	brrelex2i 5700	. 2 ⊢ (1o ≼ 𝐴 → 𝐴 ∈ V)
5		0sdomg 9071	. . 3 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
6		n0 4303	. . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
7		snssi 4741	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴)
8		df1o2 8437	. . . . . . . . . . 11 ⊢ 1o = {∅}
9		0ex 5254	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
10		vex 3457	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
11		en2sn 9015	. . . . . . . . . . . 12 ⊢ ((∅ ∈ V ∧ 𝑥 ∈ V) → {∅} ≈ {𝑥})
12	9, 10, 11	mp2an 702	. . . . . . . . . . 11 ⊢ {∅} ≈ {𝑥}
13	8, 12	eqbrtri 5118	. . . . . . . . . 10 ⊢ 1o ≈ {𝑥}
14		endom 8953	. . . . . . . . . 10 ⊢ (1o ≈ {𝑥} → 1o ≼ {𝑥})
15	13, 14	ax-mp 5	. . . . . . . . 9 ⊢ 1o ≼ {𝑥}
16		domssr 8973	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ {𝑥} ⊆ 𝐴 ∧ 1o ≼ {𝑥}) → 1o ≼ 𝐴)
17	15, 16	mp3an3 1470	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ {𝑥} ⊆ 𝐴) → 1o ≼ 𝐴)
18	17	ex 416	. . . . . . 7 ⊢ (𝐴 ∈ V → ({𝑥} ⊆ 𝐴 → 1o ≼ 𝐴))
19	7, 18	syl5 34	. . . . . 6 ⊢ (𝐴 ∈ V → (𝑥 ∈ 𝐴 → 1o ≼ 𝐴))
20	19	exlimdv 1952	. . . . 5 ⊢ (𝐴 ∈ V → (∃𝑥 𝑥 ∈ 𝐴 → 1o ≼ 𝐴))
21	6, 20	biimtrid 244	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ≠ ∅ → 1o ≼ 𝐴))
22		1n0 8449	. . . . . . 7 ⊢ 1o ≠ ∅
23		dom0 9070	. . . . . . 7 ⊢ (1o ≼ ∅ ↔ 1o = ∅)
24	22, 23	nemtbir 3052	. . . . . 6 ⊢ ¬ 1o ≼ ∅
25		breq2 5101	. . . . . 6 ⊢ (𝐴 = ∅ → (1o ≼ 𝐴 ↔ 1o ≼ ∅))
26	24, 25	mtbiri 329	. . . . 5 ⊢ (𝐴 = ∅ → ¬ 1o ≼ 𝐴)
27	26	necon2ai 2985	. . . 4 ⊢ (1o ≼ 𝐴 → 𝐴 ≠ ∅)
28	21, 27	impbid1 227	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ≠ ∅ ↔ 1o ≼ 𝐴))
29	5, 28	bitrd 281	. 2 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 1o ≼ 𝐴))
30	2, 4, 29	pm5.21nii 380	1 ⊢ (∅ ≺ 𝐴 ↔ 1o ≼ 𝐴)

Syntax hints:   ↔ wb 208   = wceq 1559  ∃wex 1798   ∈ wcel 2141   ≠ wne 2956  Vcvv 3453   ⊆ wss 3902  ∅c0 4283  {csn 4579   class class class wbr 5097  1_oc1o 8423   ≈ cen 8917   ≼ cdom 8918   ≺ csdm 8919

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-suc 6346  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-1o 8430  df-en 8921  df-dom 8922  df-sdom 8923

This theorem is referenced by:  1sdom2  9185  1sdom2dom  9191  djulepw  10142  fin45  10342  gchxpidm  10620  rankcf  10728  snct  32874