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Theorem enm 6813
Description: A set equinumerous to an inhabited set is inhabited. (Contributed by Jim Kingdon, 19-May-2020.)
Assertion
Ref Expression
enm ((𝐴𝐵 ∧ ∃𝑥 𝑥𝐴) → ∃𝑦 𝑦𝐵)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem enm
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 bren 6740 . . . . 5 (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵)
2 f1of 5456 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵𝑓:𝐴𝐵)
3 ffvelcdm 5644 . . . . . . . . 9 ((𝑓:𝐴𝐵𝑥𝐴) → (𝑓𝑥) ∈ 𝐵)
4 elex2 2753 . . . . . . . . 9 ((𝑓𝑥) ∈ 𝐵 → ∃𝑦 𝑦𝐵)
53, 4syl 14 . . . . . . . 8 ((𝑓:𝐴𝐵𝑥𝐴) → ∃𝑦 𝑦𝐵)
65ex 115 . . . . . . 7 (𝑓:𝐴𝐵 → (𝑥𝐴 → ∃𝑦 𝑦𝐵))
72, 6syl 14 . . . . . 6 (𝑓:𝐴1-1-onto𝐵 → (𝑥𝐴 → ∃𝑦 𝑦𝐵))
87exlimiv 1598 . . . . 5 (∃𝑓 𝑓:𝐴1-1-onto𝐵 → (𝑥𝐴 → ∃𝑦 𝑦𝐵))
91, 8sylbi 121 . . . 4 (𝐴𝐵 → (𝑥𝐴 → ∃𝑦 𝑦𝐵))
109com12 30 . . 3 (𝑥𝐴 → (𝐴𝐵 → ∃𝑦 𝑦𝐵))
1110exlimiv 1598 . 2 (∃𝑥 𝑥𝐴 → (𝐴𝐵 → ∃𝑦 𝑦𝐵))
1211impcom 125 1 ((𝐴𝐵 ∧ ∃𝑥 𝑥𝐴) → ∃𝑦 𝑦𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wex 1492  wcel 2148   class class class wbr 4000  wf 5207  1-1-ontowf1o 5210  cfv 5211  cen 6731
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-en 6734
This theorem is referenced by:  ssfilem  6868  diffitest  6880
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