Theorem enm 6999

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 6912	. . . . 5 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
2		f1of 5580	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴⟶𝐵)
3		ffvelcdm 5776	. . . . . . . . 9 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ 𝐵)
4		elex2 2817	. . . . . . . . 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 1644	. . . . 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 1644	. 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 1538   ∈ wcel 2200   class class class wbr 4086  ⟶wf 5320  –1-1-onto→wf1o 5323  ‘cfv 5324   ≈ cen 6902

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 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 4205  ax-pow 4262  ax-pr 4297  ax-un 4528

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 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-en 6905

This theorem is referenced by:  ssfilem  7057  diffitest  7069