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Theorem gchinf 10081

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 10080	. . 3 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (𝐴 ⊔ 1_o) ≈ 𝐴)
2	1	ensymd 8562	. 2 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → 𝐴 ≈ (𝐴 ⊔ 1_o))
3		isfin4-2 9738	. . . 4 ⊢ (𝐴 ∈ GCH → (𝐴 ∈ Fin^IV ↔ ¬ ω ≼ 𝐴))
4	3	adantr 483	. . 3 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (𝐴 ∈ Fin^IV ↔ ¬ ω ≼ 𝐴))
5		isfin4p1 9739	. . . 4 ⊢ (𝐴 ∈ Fin^IV ↔ 𝐴 ≺ (𝐴 ⊔ 1_o))
6		sdomnen 8540	. . . 4 ⊢ (𝐴 ≺ (𝐴 ⊔ 1_o) → ¬ 𝐴 ≈ (𝐴 ⊔ 1_o))
7	5, 6	sylbi 219	. . 3 ⊢ (𝐴 ∈ Fin^IV → ¬ 𝐴 ≈ (𝐴 ⊔ 1_o))
8	4, 7	syl6bir 256	. 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 208 ∧ wa 398 ∈ wcel 2114 class class class wbr 5068 ωcom 7582 1_oc1o 8097 ≈ cen 8508 ≼ cdom 8509 ≺ csdm 8510 Fincfn 8511 ⊔ cdju 9329 Fin^IVcfin4 9704 GCHcgch 10044

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-oi 8976 df-dju 9332 df-card 9370 df-fin4 9711 df-gch 10045

This theorem is referenced by: gchdjuidm 10092 gchxpidm 10093 gchina 10123

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