Metamath Proof Explorer


Theorem rex2dom

Description: A set that has at least 2 different members dominates ordinal 2. (Contributed by BTernaryTau, 30-Dec-2024)

Ref Expression
Assertion rex2dom ( ( 𝐴𝑉 ∧ ∃ 𝑥𝐴𝑦𝐴 𝑥𝑦 ) → 2o𝐴 )

Proof

Step Hyp Ref Expression
1 elex ( 𝐴𝑉𝐴 ∈ V )
2 prssi ( ( 𝑥𝐴𝑦𝐴 ) → { 𝑥 , 𝑦 } ⊆ 𝐴 )
3 df2o3 2o = { ∅ , 1o }
4 0ex ∅ ∈ V
5 4 a1i ( 𝑥𝑦 → ∅ ∈ V )
6 1oex 1o ∈ V
7 6 a1i ( 𝑥𝑦 → 1o ∈ V )
8 vex 𝑥 ∈ V
9 8 a1i ( 𝑥𝑦𝑥 ∈ V )
10 vex 𝑦 ∈ V
11 10 a1i ( 𝑥𝑦𝑦 ∈ V )
12 1n0 1o ≠ ∅
13 12 necomi ∅ ≠ 1o
14 13 a1i ( 𝑥𝑦 → ∅ ≠ 1o )
15 id ( 𝑥𝑦𝑥𝑦 )
16 5 7 9 11 14 15 en2prd ( 𝑥𝑦 → { ∅ , 1o } ≈ { 𝑥 , 𝑦 } )
17 3 16 eqbrtrid ( 𝑥𝑦 → 2o ≈ { 𝑥 , 𝑦 } )
18 endom ( 2o ≈ { 𝑥 , 𝑦 } → 2o ≼ { 𝑥 , 𝑦 } )
19 17 18 syl ( 𝑥𝑦 → 2o ≼ { 𝑥 , 𝑦 } )
20 domssr ( ( 𝐴 ∈ V ∧ { 𝑥 , 𝑦 } ⊆ 𝐴 ∧ 2o ≼ { 𝑥 , 𝑦 } ) → 2o𝐴 )
21 20 3expib ( 𝐴 ∈ V → ( ( { 𝑥 , 𝑦 } ⊆ 𝐴 ∧ 2o ≼ { 𝑥 , 𝑦 } ) → 2o𝐴 ) )
22 2 19 21 syl2ani ( 𝐴 ∈ V → ( ( ( 𝑥𝐴𝑦𝐴 ) ∧ 𝑥𝑦 ) → 2o𝐴 ) )
23 22 expd ( 𝐴 ∈ V → ( ( 𝑥𝐴𝑦𝐴 ) → ( 𝑥𝑦 → 2o𝐴 ) ) )
24 23 rexlimdvv ( 𝐴 ∈ V → ( ∃ 𝑥𝐴𝑦𝐴 𝑥𝑦 → 2o𝐴 ) )
25 1 24 syl ( 𝐴𝑉 → ( ∃ 𝑥𝐴𝑦𝐴 𝑥𝑦 → 2o𝐴 ) )
26 25 imp ( ( 𝐴𝑉 ∧ ∃ 𝑥𝐴𝑦𝐴 𝑥𝑦 ) → 2o𝐴 )