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Theorem gchen1 9639
Description: If 𝐴𝐵 < 𝒫 𝐴, and 𝐴 is an infinite GCH-set, then 𝐴 = 𝐵 in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
gchen1 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≺ 𝒫 𝐴)) → 𝐴𝐵)

Proof of Theorem gchen1
StepHypRef Expression
1 simprl 811 . 2 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≺ 𝒫 𝐴)) → 𝐴𝐵)
2 gchi 9638 . . . . . . 7 ((𝐴 ∈ GCH ∧ 𝐴𝐵𝐵 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin)
323com23 1121 . . . . . 6 ((𝐴 ∈ GCH ∧ 𝐵 ≺ 𝒫 𝐴𝐴𝐵) → 𝐴 ∈ Fin)
433expia 1115 . . . . 5 ((𝐴 ∈ GCH ∧ 𝐵 ≺ 𝒫 𝐴) → (𝐴𝐵𝐴 ∈ Fin))
54con3dimp 456 . . . 4 (((𝐴 ∈ GCH ∧ 𝐵 ≺ 𝒫 𝐴) ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐴𝐵)
65an32s 881 . . 3 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ 𝐵 ≺ 𝒫 𝐴) → ¬ 𝐴𝐵)
76adantrl 754 . 2 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≺ 𝒫 𝐴)) → ¬ 𝐴𝐵)
8 bren2 8152 . 2 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵))
91, 7, 8sylanbrc 701 1 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≺ 𝒫 𝐴)) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wcel 2139  𝒫 cpw 4302   class class class wbr 4804  cen 8118  cdom 8119  csdm 8120  Fincfn 8121  GCHcgch 9634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-f1o 6056  df-en 8122  df-dom 8123  df-sdom 8124  df-gch 9635
This theorem is referenced by:  gchor  9641  gchcda1  9670  gchcdaidm  9682  gchxpidm  9683  gchhar  9693
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