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Theorem gchor 10668
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 772 . . 3 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≼ 𝒫 𝐴)) → 𝐵 ≼ 𝒫 𝐴)
2 brdom2 9023 . . 3 (𝐵 ≼ 𝒫 𝐴 ↔ (𝐵 ≺ 𝒫 𝐴𝐵 ≈ 𝒫 𝐴))
31, 2sylib 218 . 2 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≼ 𝒫 𝐴)) → (𝐵 ≺ 𝒫 𝐴𝐵 ≈ 𝒫 𝐴))
4 gchen1 10666 . . . . 5 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≺ 𝒫 𝐴)) → 𝐴𝐵)
54expr 456 . . . 4 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ 𝐴𝐵) → (𝐵 ≺ 𝒫 𝐴𝐴𝐵))
65adantrr 717 . . 3 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≼ 𝒫 𝐴)) → (𝐵 ≺ 𝒫 𝐴𝐴𝐵))
76orim1d 967 . 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  wa 395  wo 847  wcel 2107  𝒫 cpw 4599   class class class wbr 5142  cen 8983  cdom 8984  csdm 8985  Fincfn 8986  GCHcgch 10661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-f1o 6567  df-en 8987  df-dom 8988  df-sdom 8989  df-gch 10662
This theorem is referenced by:  gchdomtri  10670  gchpwdom  10711
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