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Theorem gchinf 10616

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 10615	. . 3 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (𝐴 ⊔ 1_o) ≈ 𝐴)
2	1	ensymd 8978	. 2 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → 𝐴 ≈ (𝐴 ⊔ 1_o))
3		isfin4-2 10273	. . . 4 ⊢ (𝐴 ∈ GCH → (𝐴 ∈ Fin^IV ↔ ¬ ω ≼ 𝐴))
4	3	adantr 480	. . 3 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (𝐴 ∈ Fin^IV ↔ ¬ ω ≼ 𝐴))
5		isfin4p1 10274	. . . 4 ⊢ (𝐴 ∈ Fin^IV ↔ 𝐴 ≺ (𝐴 ⊔ 1_o))
6		sdomnen 8954	. . . 4 ⊢ (𝐴 ≺ (𝐴 ⊔ 1_o) → ¬ 𝐴 ≈ (𝐴 ⊔ 1_o))
7	5, 6	sylbi 217	. . 3 ⊢ (𝐴 ∈ Fin^IV → ¬ 𝐴 ≈ (𝐴 ⊔ 1_o))
8	4, 7	biimtrrdi 254	. 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 206 ∧ wa 395 ∈ wcel 2109 class class class wbr 5109 ωcom 7844 1_oc1o 8429 ≈ cen 8917 ≼ cdom 8918 ≺ csdm 8919 Fincfn 8920 ⊔ cdju 9857 Fin^IVcfin4 10239 GCHcgch 10579

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-inf2 9600

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-int 4913 df-iun 4959 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-se 5594 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-isom 6522 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-om 7845 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-1o 8436 df-2o 8437 df-er 8673 df-map 8803 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-oi 9469 df-dju 9860 df-card 9898 df-fin4 10246 df-gch 10580

This theorem is referenced by: gchdjuidm 10627 gchxpidm 10628 gchina 10658

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