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Theorem enm 6682
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 6609 . . . . 5 (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵)
2 f1of 5335 . . . . . . 7 (𝑓:𝐴1-1-onto𝐵𝑓:𝐴𝐵)
3 ffvelrn 5521 . . . . . . . . 9 ((𝑓:𝐴𝐵𝑥𝐴) → (𝑓𝑥) ∈ 𝐵)
4 elex2 2676 . . . . . . . . 9 ((𝑓𝑥) ∈ 𝐵 → ∃𝑦 𝑦𝐵)
53, 4syl 14 . . . . . . . 8 ((𝑓:𝐴𝐵𝑥𝐴) → ∃𝑦 𝑦𝐵)
65ex 114 . . . . . . 7 (𝑓:𝐴𝐵 → (𝑥𝐴 → ∃𝑦 𝑦𝐵))
72, 6syl 14 . . . . . 6 (𝑓:𝐴1-1-onto𝐵 → (𝑥𝐴 → ∃𝑦 𝑦𝐵))
87exlimiv 1562 . . . . 5 (∃𝑓 𝑓:𝐴1-1-onto𝐵 → (𝑥𝐴 → ∃𝑦 𝑦𝐵))
91, 8sylbi 120 . . . 4 (𝐴𝐵 → (𝑥𝐴 → ∃𝑦 𝑦𝐵))
109com12 30 . . 3 (𝑥𝐴 → (𝐴𝐵 → ∃𝑦 𝑦𝐵))
1110exlimiv 1562 . 2 (∃𝑥 𝑥𝐴 → (𝐴𝐵 → ∃𝑦 𝑦𝐵))
1211impcom 124 1 ((𝐴𝐵 ∧ ∃𝑥 𝑥𝐴) → ∃𝑦 𝑦𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wex 1453  wcel 1465   class class class wbr 3899  wf 5089  1-1-ontowf1o 5092  cfv 5093  cen 6600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-en 6603
This theorem is referenced by:  ssfilem  6737  diffitest  6749
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