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Theorem infdif2 9609
Description: Cardinality ordering for an infinite class difference. (Contributed by NM, 24-Mar-2007.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
infdif2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ((𝐴𝐵) ≼ 𝐵𝐴𝐵))

Proof of Theorem infdif2
StepHypRef Expression
1 domnsym 8619 . . . . . . 7 ((𝐴𝐵) ≼ 𝐵 → ¬ 𝐵 ≺ (𝐴𝐵))
2 simp3 1135 . . . . . . . 8 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵𝐴)
3 infdif 9608 . . . . . . . . 9 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≈ 𝐴)
43ensymd 8535 . . . . . . . 8 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐴 ≈ (𝐴𝐵))
5 sdomentr 8627 . . . . . . . 8 ((𝐵𝐴𝐴 ≈ (𝐴𝐵)) → 𝐵 ≺ (𝐴𝐵))
62, 4, 5syl2anc 587 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → 𝐵 ≺ (𝐴𝐵))
71, 6nsyl3 140 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → ¬ (𝐴𝐵) ≼ 𝐵)
873expia 1118 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐵𝐴 → ¬ (𝐴𝐵) ≼ 𝐵))
983adant2 1128 . . . 4 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐵𝐴 → ¬ (𝐴𝐵) ≼ 𝐵))
109con2d 136 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ((𝐴𝐵) ≼ 𝐵 → ¬ 𝐵𝐴))
11 domtri2 9394 . . . 4 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
12113adant3 1129 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
1310, 12sylibrd 262 . 2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ((𝐴𝐵) ≼ 𝐵𝐴𝐵))
14 simp1 1133 . . . 4 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ∈ dom card)
15 difss 4084 . . . 4 (𝐴𝐵) ⊆ 𝐴
16 ssdomg 8530 . . . 4 (𝐴 ∈ dom card → ((𝐴𝐵) ⊆ 𝐴 → (𝐴𝐵) ≼ 𝐴))
1714, 15, 16mpisyl 21 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴𝐵) ≼ 𝐴)
18 domtr 8537 . . . 4 (((𝐴𝐵) ≼ 𝐴𝐴𝐵) → (𝐴𝐵) ≼ 𝐵)
1918ex 416 . . 3 ((𝐴𝐵) ≼ 𝐴 → (𝐴𝐵 → (𝐴𝐵) ≼ 𝐵))
2017, 19syl 17 . 2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴𝐵 → (𝐴𝐵) ≼ 𝐵))
2113, 20impbid 215 1 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ((𝐴𝐵) ≼ 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  w3a 1084  wcel 2115  cdif 3907  wss 3910   class class class wbr 5039  dom cdm 5528  ωcom 7555  cen 8481  cdom 8482  csdm 8483  cardccrd 9340
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-inf2 9080
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-se 5488  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-2o 8078  df-oadd 8081  df-er 8264  df-en 8485  df-dom 8486  df-sdom 8487  df-fin 8488  df-oi 8950  df-dju 9306  df-card 9344
This theorem is referenced by:  axgroth3  10230
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