Theorem 1dom1el 6992

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 6919	. . 3 ⊢ (𝐴 ≼ 1o → ∃𝑓 𝑓:𝐴–1-1→1o)
2	1	3ad2ant1 1044	. 2 ⊢ ((𝐴 ≼ 1o ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ∃𝑓 𝑓:𝐴–1-1→1o)
3		f1f 5542	. . . . . . 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 1027	. . . . . 6 ⊢ (((𝐴 ≼ 1o ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ 𝑓:𝐴–1-1→1o) → 𝐵 ∈ 𝐴)
6	4, 5	ffvelcdmd 5783	. . . . 5 ⊢ (((𝐴 ≼ 1o ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ 𝑓:𝐴–1-1→1o) → (𝑓‘𝐵) ∈ 1o)
7		el1o 6604	. . . . 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 1028	. . . . . 6 ⊢ (((𝐴 ≼ 1o ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ 𝑓:𝐴–1-1→1o) → 𝐶 ∈ 𝐴)
10	4, 9	ffvelcdmd 5783	. . . . 5 ⊢ (((𝐴 ≼ 1o ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ 𝑓:𝐴–1-1→1o) → (𝑓‘𝐶) ∈ 1o)
11		el1o 6604	. . . . 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 2267	. . 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 5909	. . . 4 ⊢ ((𝑓:𝐴–1-1→1o ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ((𝑓‘𝐵) = (𝑓‘𝐶) → 𝐵 = 𝐶))
16	14, 5, 9, 15	syl12anc 1271	. . 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 1947	1 ⊢ ((𝐴 ≼ 1o ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → 𝐵 = 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1004   = wceq 1397  ∃wex 1540   ∈ wcel 2202  ∅c0 3494   class class class wbr 4088  ⟶wf 5322  –1-1→wf1 5323  ‘cfv 5326  1_oc1o 6574   ≼ cdom 6907

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fv 5334  df-1o 6581  df-dom 6910

This theorem is referenced by:  modom  6993