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Theorem gchor 9612
Description: If 𝐴𝐵 ≤ 𝒫 𝐴, and 𝐴 is an infinite GCH-set, then either 𝐴 = 𝐵 or 𝐵 = 𝒫 𝐴 in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
gchor (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≼ 𝒫 𝐴)) → (𝐴𝐵𝐵 ≈ 𝒫 𝐴))

Proof of Theorem gchor
StepHypRef Expression
1 simprr 813 . . 3 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≼ 𝒫 𝐴)) → 𝐵 ≼ 𝒫 𝐴)
2 brdom2 8139 . . 3 (𝐵 ≼ 𝒫 𝐴 ↔ (𝐵 ≺ 𝒫 𝐴𝐵 ≈ 𝒫 𝐴))
31, 2sylib 208 . 2 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≼ 𝒫 𝐴)) → (𝐵 ≺ 𝒫 𝐴𝐵 ≈ 𝒫 𝐴))
4 gchen1 9610 . . . . 5 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≺ 𝒫 𝐴)) → 𝐴𝐵)
54expr 644 . . . 4 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ 𝐴𝐵) → (𝐵 ≺ 𝒫 𝐴𝐴𝐵))
65adantrr 755 . . 3 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≼ 𝒫 𝐴)) → (𝐵 ≺ 𝒫 𝐴𝐴𝐵))
76orim1d 920 . 2 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≼ 𝒫 𝐴)) → ((𝐵 ≺ 𝒫 𝐴𝐵 ≈ 𝒫 𝐴) → (𝐴𝐵𝐵 ≈ 𝒫 𝐴)))
83, 7mpd 15 1 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≼ 𝒫 𝐴)) → (𝐴𝐵𝐵 ≈ 𝒫 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383  wcel 2127  𝒫 cpw 4290   class class class wbr 4792  cen 8106  cdom 8107  csdm 8108  Fincfn 8109  GCHcgch 9605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-br 4793  df-opab 4853  df-xp 5260  df-rel 5261  df-f1o 6044  df-en 8110  df-dom 8111  df-sdom 8112  df-gch 9606
This theorem is referenced by:  gchdomtri  9614  gchpwdom  9655
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