gchinf - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > gchinf		Structured version Visualization version GIF version

Theorem gchinf 10648

Description: An infinite GCH-set is Dedekind-infinite. (Contributed by Mario Carneiro, 31-May-2015.)

**Assertion**
Ref	Expression
gchinf	⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → ω ≼ 𝐴)

**Proof of Theorem gchinf**
Step	Hyp	Ref	Expression
1		gchdju1 10647	. . 3 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (𝐴 ⊔ 1_o) ≈ 𝐴)
2	1	ensymd 8997	. 2 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → 𝐴 ≈ (𝐴 ⊔ 1_o))
3		isfin4-2 10305	. . . 4 ⊢ (𝐴 ∈ GCH → (𝐴 ∈ Fin^IV ↔ ¬ ω ≼ 𝐴))
4	3	adantr 481	. . 3 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (𝐴 ∈ Fin^IV ↔ ¬ ω ≼ 𝐴))
5		isfin4p1 10306	. . . 4 ⊢ (𝐴 ∈ Fin^IV ↔ 𝐴 ≺ (𝐴 ⊔ 1_o))
6		sdomnen 8973	. . . 4 ⊢ (𝐴 ≺ (𝐴 ⊔ 1_o) → ¬ 𝐴 ≈ (𝐴 ⊔ 1_o))
7	5, 6	sylbi 216	. . 3 ⊢ (𝐴 ∈ Fin^IV → ¬ 𝐴 ≈ (𝐴 ⊔ 1_o))
8	4, 7	syl6bir 253	. 2 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (¬ ω ≼ 𝐴 → ¬ 𝐴 ≈ (𝐴 ⊔ 1_o)))
9	2, 8	mt4d 117	1 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → ω ≼ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 class class class wbr 5147 ωcom 7851 1_oc1o 8455 ≈ cen 8932 ≼ cdom 8933 ≺ csdm 8934 Fincfn 8935 ⊔ cdju 9889 Fin^IVcfin4 10271 GCHcgch 10611

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-oi 9501 df-dju 9892 df-card 9930 df-fin4 10278 df-gch 10612

This theorem is referenced by: gchdjuidm 10659 gchxpidm 10660 gchina 10690

W3C validator