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Theorem 0sdom1dom 9261
Description: Strict dominance over 0 is the same as dominance over 1. For a shorter proof requiring ax-un 7738, see 0sdom1domALT . (Contributed by NM, 28-Sep-2004.) Avoid ax-un 7738. (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 8969 . . 3 Rel ≺
21brrelex2i 5729 . 2 (∅ ≺ 𝐴𝐴 ∈ V)
3 reldom 8968 . . 3 Rel ≼
43brrelex2i 5729 . 2 (1o𝐴𝐴 ∈ V)
5 0sdomg 9127 . . 3 (𝐴 ∈ V → (∅ ≺ 𝐴𝐴 ≠ ∅))
6 n0 4342 . . . . 5 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
7 snssi 4807 . . . . . . 7 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
8 df1o2 8492 . . . . . . . . . . 11 1o = {∅}
9 0ex 5302 . . . . . . . . . . . 12 ∅ ∈ V
10 vex 3467 . . . . . . . . . . . 12 𝑥 ∈ V
11 en2sn 9064 . . . . . . . . . . . 12 ((∅ ∈ V ∧ 𝑥 ∈ V) → {∅} ≈ {𝑥})
129, 10, 11mp2an 690 . . . . . . . . . . 11 {∅} ≈ {𝑥}
138, 12eqbrtri 5164 . . . . . . . . . 10 1o ≈ {𝑥}
14 endom 8998 . . . . . . . . . 10 (1o ≈ {𝑥} → 1o ≼ {𝑥})
1513, 14ax-mp 5 . . . . . . . . 9 1o ≼ {𝑥}
16 domssr 9018 . . . . . . . . 9 ((𝐴 ∈ V ∧ {𝑥} ⊆ 𝐴 ∧ 1o ≼ {𝑥}) → 1o𝐴)
1715, 16mp3an3 1446 . . . . . . . 8 ((𝐴 ∈ V ∧ {𝑥} ⊆ 𝐴) → 1o𝐴)
1817ex 411 . . . . . . 7 (𝐴 ∈ V → ({𝑥} ⊆ 𝐴 → 1o𝐴))
197, 18syl5 34 . . . . . 6 (𝐴 ∈ V → (𝑥𝐴 → 1o𝐴))
2019exlimdv 1928 . . . . 5 (𝐴 ∈ V → (∃𝑥 𝑥𝐴 → 1o𝐴))
216, 20biimtrid 241 . . . 4 (𝐴 ∈ V → (𝐴 ≠ ∅ → 1o𝐴))
22 1n0 8507 . . . . . . 7 1o ≠ ∅
23 dom0 9125 . . . . . . 7 (1o ≼ ∅ ↔ 1o = ∅)
2422, 23nemtbir 3028 . . . . . 6 ¬ 1o ≼ ∅
25 breq2 5147 . . . . . 6 (𝐴 = ∅ → (1o𝐴 ↔ 1o ≼ ∅))
2624, 25mtbiri 326 . . . . 5 (𝐴 = ∅ → ¬ 1o𝐴)
2726necon2ai 2960 . . . 4 (1o𝐴𝐴 ≠ ∅)
2821, 27impbid1 224 . . 3 (𝐴 ∈ V → (𝐴 ≠ ∅ ↔ 1o𝐴))
295, 28bitrd 278 . 2 (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 1o𝐴))
302, 4, 29pm5.21nii 377 1 (∅ ≺ 𝐴 ↔ 1o𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  wex 1773  wcel 2098  wne 2930  Vcvv 3463  wss 3939  c0 4318  {csn 4624   class class class wbr 5143  1oc1o 8478  cen 8959  cdom 8960  csdm 8961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-suc 6370  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-1o 8485  df-en 8963  df-dom 8964  df-sdom 8965
This theorem is referenced by:  1sdom2  9263  sdom1OLD  9266  1sdom2dom  9270  djulepw  10215  fin45  10415  gchxpidm  10692  rankcf  10800  snct  32540
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