Theorem 1sdom 9144

Description: A set that strictly dominates ordinal 1 has at least 2 different members. (Closely related to 2dom 8955.) (Contributed by Mario Carneiro, 12-Jan-2013.) Avoid ax-un 7671. (Revised by BTernaryTau, 30-Dec-2024.)

Assertion

Ref	Expression
1sdom	⊢ (𝐴 ∈ 𝑉 → (1o ≺ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦))

Distinct variable group:   𝑥,𝐴,𝑦

Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem 1sdom

Step	Hyp	Ref	Expression
1		1sdom2dom 9143	. 2 ⊢ (1o ≺ 𝐴 ↔ 2o ≼ 𝐴)
2		2dom 8955	. . 3 ⊢ (2o ≼ 𝐴 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦)
3		df-ne 2926	. . . . . 6 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦)
4	3	2rexbii 3105	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦)
5		rex2dom 9142	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦) → 2o ≼ 𝐴)
6	4, 5	sylan2br 595	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦) → 2o ≼ 𝐴)
7	6	ex 412	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦 → 2o ≼ 𝐴))
8	2, 7	impbid2 226	. 2 ⊢ (𝐴 ∈ 𝑉 → (2o ≼ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦))
9	1, 8	bitrid 283	1 ⊢ (𝐴 ∈ 𝑉 → (1o ≺ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053   class class class wbr 5092  1_oc1o 8381  2_oc2o 8382   ≼ cdom 8870   ≺ csdm 8871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-1o 8388  df-2o 8389  df-en 8873  df-dom 8874  df-sdom 8875

This theorem is referenced by:  unxpdomlem3  9147  frgpnabl  19754  isnzr2  20403