Theorem enm 7071

Description: A set equinumerous to an inhabited set is inhabited. (Contributed by Jim Kingdon, 19-May-2020.)

Assertion

Ref	Expression
enm	⊢ ((𝐴 ≈ 𝐵 ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∃𝑦 𝑦 ∈ 𝐵)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem enm

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren 6983	. . . . 5 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
2		f1of 5614	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴⟶𝐵)
3		ffvelcdm 5810	. . . . . . . . 9 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ 𝐵)
4		elex2 2830	. . . . . . . . 9 ⊢ ((𝑓‘𝑥) ∈ 𝐵 → ∃𝑦 𝑦 ∈ 𝐵)
5	3, 4	syl 14	. . . . . . . 8 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 𝑦 ∈ 𝐵)
6	5	ex 115	. . . . . . 7 ⊢ (𝑓:𝐴⟶𝐵 → (𝑥 ∈ 𝐴 → ∃𝑦 𝑦 ∈ 𝐵))
7	2, 6	syl 14	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (𝑥 ∈ 𝐴 → ∃𝑦 𝑦 ∈ 𝐵))
8	7	exlimiv 1647	. . . . 5 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (𝑥 ∈ 𝐴 → ∃𝑦 𝑦 ∈ 𝐵))
9	1, 8	sylbi 121	. . . 4 ⊢ (𝐴 ≈ 𝐵 → (𝑥 ∈ 𝐴 → ∃𝑦 𝑦 ∈ 𝐵))
10	9	com12 30	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐴 ≈ 𝐵 → ∃𝑦 𝑦 ∈ 𝐵))
11	10	exlimiv 1647	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐴 ≈ 𝐵 → ∃𝑦 𝑦 ∈ 𝐵))
12	11	impcom 125	1 ⊢ ((𝐴 ≈ 𝐵 ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∃𝑦 𝑦 ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 104  ∃wex 1541   ∈ wcel 2203   class class class wbr 4109  ⟶wf 5348  –1-1-onto→wf1o 5351  ‘cfv 5352   ≈ cen 6973

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-en 6976

This theorem is referenced by:  ssfilem  7130  ssfilemd  7132  diffitest  7144