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Theorem 0sdom1dom 9189
Description: Strict dominance over 0 is the same as dominance over 1. For a shorter proof requiring ax-un 7677, see 0sdom1domALT . (Contributed by NM, 28-Sep-2004.) Avoid ax-un 7677. (Revised by BTernaryTau, 7-Dec-2024.)
Assertion
Ref Expression
0sdom1dom (∅ ≺ 𝐴 ↔ 1o𝐴)

Proof of Theorem 0sdom1dom
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 relsdom 8897 . . 3 Rel ≺
21brrelex2i 5694 . 2 (∅ ≺ 𝐴𝐴 ∈ V)
3 reldom 8896 . . 3 Rel ≼
43brrelex2i 5694 . 2 (1o𝐴𝐴 ∈ V)
5 0sdomg 9055 . . 3 (𝐴 ∈ V → (∅ ≺ 𝐴𝐴 ≠ ∅))
6 n0 4311 . . . . 5 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
7 snssi 4773 . . . . . . 7 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
8 df1o2 8424 . . . . . . . . . . 11 1o = {∅}
9 0ex 5269 . . . . . . . . . . . 12 ∅ ∈ V
10 vex 3452 . . . . . . . . . . . 12 𝑥 ∈ V
11 en2sn 8992 . . . . . . . . . . . 12 ((∅ ∈ V ∧ 𝑥 ∈ V) → {∅} ≈ {𝑥})
129, 10, 11mp2an 691 . . . . . . . . . . 11 {∅} ≈ {𝑥}
138, 12eqbrtri 5131 . . . . . . . . . 10 1o ≈ {𝑥}
14 endom 8926 . . . . . . . . . 10 (1o ≈ {𝑥} → 1o ≼ {𝑥})
1513, 14ax-mp 5 . . . . . . . . 9 1o ≼ {𝑥}
16 domssr 8946 . . . . . . . . 9 ((𝐴 ∈ V ∧ {𝑥} ⊆ 𝐴 ∧ 1o ≼ {𝑥}) → 1o𝐴)
1715, 16mp3an3 1451 . . . . . . . 8 ((𝐴 ∈ V ∧ {𝑥} ⊆ 𝐴) → 1o𝐴)
1817ex 414 . . . . . . 7 (𝐴 ∈ V → ({𝑥} ⊆ 𝐴 → 1o𝐴))
197, 18syl5 34 . . . . . 6 (𝐴 ∈ V → (𝑥𝐴 → 1o𝐴))
2019exlimdv 1937 . . . . 5 (𝐴 ∈ V → (∃𝑥 𝑥𝐴 → 1o𝐴))
216, 20biimtrid 241 . . . 4 (𝐴 ∈ V → (𝐴 ≠ ∅ → 1o𝐴))
22 1n0 8439 . . . . . . 7 1o ≠ ∅
23 dom0 9053 . . . . . . 7 (1o ≼ ∅ ↔ 1o = ∅)
2422, 23nemtbir 3041 . . . . . 6 ¬ 1o ≼ ∅
25 breq2 5114 . . . . . 6 (𝐴 = ∅ → (1o𝐴 ↔ 1o ≼ ∅))
2624, 25mtbiri 327 . . . . 5 (𝐴 = ∅ → ¬ 1o𝐴)
2726necon2ai 2974 . . . 4 (1o𝐴𝐴 ≠ ∅)
2821, 27impbid1 224 . . 3 (𝐴 ∈ V → (𝐴 ≠ ∅ ↔ 1o𝐴))
295, 28bitrd 279 . 2 (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 1o𝐴))
302, 4, 29pm5.21nii 380 1 (∅ ≺ 𝐴 ↔ 1o𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  wex 1782  wcel 2107  wne 2944  Vcvv 3448  wss 3915  c0 4287  {csn 4591   class class class wbr 5110  1oc1o 8410  cen 8887  cdom 8888  csdm 8889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-suc 6328  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-1o 8417  df-en 8891  df-dom 8892  df-sdom 8893
This theorem is referenced by:  1sdom2  9191  sdom1OLD  9194  1sdom2dom  9198  djulepw  10135  fin45  10335  gchxpidm  10612  rankcf  10720  snct  31672
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