Theorem 1dom1el 16312

Description: If a set is dominated by one, then any two of its elements are equal. (Contributed by Jim Kingdon, 23-Apr-2025.)

Assertion

Ref	Expression
1dom1el	⊢ ((𝐴 ≼ 1o ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → 𝐵 = 𝐶)

Proof of Theorem 1dom1el

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brdomi 6896	. . 3 ⊢ (𝐴 ≼ 1o → ∃𝑓 𝑓:𝐴–1-1→1o)
2	1	3ad2ant1 1042	. 2 ⊢ ((𝐴 ≼ 1o ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ∃𝑓 𝑓:𝐴–1-1→1o)
3		f1f 5530	. . . . . . 7 ⊢ (𝑓:𝐴–1-1→1o → 𝑓:𝐴⟶1o)
4	3	adantl 277	. . . . . 6 ⊢ (((𝐴 ≼ 1o ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ 𝑓:𝐴–1-1→1o) → 𝑓:𝐴⟶1o)
5		simpl2 1025	. . . . . 6 ⊢ (((𝐴 ≼ 1o ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ 𝑓:𝐴–1-1→1o) → 𝐵 ∈ 𝐴)
6	4, 5	ffvelcdmd 5770	. . . . 5 ⊢ (((𝐴 ≼ 1o ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ 𝑓:𝐴–1-1→1o) → (𝑓‘𝐵) ∈ 1o)
7		el1o 6581	. . . . 5 ⊢ ((𝑓‘𝐵) ∈ 1o ↔ (𝑓‘𝐵) = ∅)
8	6, 7	sylib 122	. . . 4 ⊢ (((𝐴 ≼ 1o ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ 𝑓:𝐴–1-1→1o) → (𝑓‘𝐵) = ∅)
9		simpl3 1026	. . . . . 6 ⊢ (((𝐴 ≼ 1o ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ 𝑓:𝐴–1-1→1o) → 𝐶 ∈ 𝐴)
10	4, 9	ffvelcdmd 5770	. . . . 5 ⊢ (((𝐴 ≼ 1o ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ 𝑓:𝐴–1-1→1o) → (𝑓‘𝐶) ∈ 1o)
11		el1o 6581	. . . . 5 ⊢ ((𝑓‘𝐶) ∈ 1o ↔ (𝑓‘𝐶) = ∅)
12	10, 11	sylib 122	. . . 4 ⊢ (((𝐴 ≼ 1o ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ 𝑓:𝐴–1-1→1o) → (𝑓‘𝐶) = ∅)
13	8, 12	eqtr4d 2265	. . 3 ⊢ (((𝐴 ≼ 1o ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ 𝑓:𝐴–1-1→1o) → (𝑓‘𝐵) = (𝑓‘𝐶))
14		simpr 110	. . . 4 ⊢ (((𝐴 ≼ 1o ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ 𝑓:𝐴–1-1→1o) → 𝑓:𝐴–1-1→1o)
15		f1veqaeq 5892	. . . 4 ⊢ ((𝑓:𝐴–1-1→1o ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ((𝑓‘𝐵) = (𝑓‘𝐶) → 𝐵 = 𝐶))
16	14, 5, 9, 15	syl12anc 1269	. . 3 ⊢ (((𝐴 ≼ 1o ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ 𝑓:𝐴–1-1→1o) → ((𝑓‘𝐵) = (𝑓‘𝐶) → 𝐵 = 𝐶))
17	13, 16	mpd 13	. 2 ⊢ (((𝐴 ≼ 1o ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ 𝑓:𝐴–1-1→1o) → 𝐵 = 𝐶)
18	2, 17	exlimddv 1945	1 ⊢ ((𝐴 ≼ 1o ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → 𝐵 = 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1002   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∅c0 3491   class class class wbr 4082  ⟶wf 5313  –1-1→wf1 5314  ‘cfv 5317  1_oc1o 6553   ≼ cdom 6884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-id 4383  df-suc 4461  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fv 5325  df-1o 6560  df-dom 6887

This theorem is referenced by: (None)