Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  carddomi2 Structured version   Visualization version   GIF version

Theorem carddomi2 8986
 Description: Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 9568, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
carddomi2 ((𝐴 ∈ dom card ∧ 𝐵𝑉) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴𝐵))

Proof of Theorem carddomi2
StepHypRef Expression
1 cardnueq0 8980 . . . . . 6 (𝐴 ∈ dom card → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
21adantr 472 . . . . 5 ((𝐴 ∈ dom card ∧ 𝐵𝑉) → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
32biimpa 502 . . . 4 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ (card‘𝐴) = ∅) → 𝐴 = ∅)
4 0domg 8252 . . . . 5 (𝐵𝑉 → ∅ ≼ 𝐵)
54ad2antlr 765 . . . 4 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ (card‘𝐴) = ∅) → ∅ ≼ 𝐵)
63, 5eqbrtrd 4826 . . 3 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ (card‘𝐴) = ∅) → 𝐴𝐵)
76a1d 25 . 2 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ (card‘𝐴) = ∅) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴𝐵))
8 fvex 6362 . . . . 5 (card‘𝐵) ∈ V
9 simprr 813 . . . . 5 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ⊆ (card‘𝐵))
10 ssdomg 8167 . . . . 5 ((card‘𝐵) ∈ V → ((card‘𝐴) ⊆ (card‘𝐵) → (card‘𝐴) ≼ (card‘𝐵)))
118, 9, 10mpsyl 68 . . . 4 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ≼ (card‘𝐵))
12 cardid2 8969 . . . . . 6 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
1312ad2antrr 764 . . . . 5 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ≈ 𝐴)
14 simprl 811 . . . . . . 7 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐴) ≠ ∅)
15 ssn0 4119 . . . . . . 7 (((card‘𝐴) ⊆ (card‘𝐵) ∧ (card‘𝐴) ≠ ∅) → (card‘𝐵) ≠ ∅)
169, 14, 15syl2anc 696 . . . . . 6 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐵) ≠ ∅)
17 ndmfv 6379 . . . . . . 7 𝐵 ∈ dom card → (card‘𝐵) = ∅)
1817necon1ai 2959 . . . . . 6 ((card‘𝐵) ≠ ∅ → 𝐵 ∈ dom card)
19 cardid2 8969 . . . . . 6 (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
2016, 18, 193syl 18 . . . . 5 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → (card‘𝐵) ≈ 𝐵)
21 domen1 8267 . . . . . 6 ((card‘𝐴) ≈ 𝐴 → ((card‘𝐴) ≼ (card‘𝐵) ↔ 𝐴 ≼ (card‘𝐵)))
22 domen2 8268 . . . . . 6 ((card‘𝐵) ≈ 𝐵 → (𝐴 ≼ (card‘𝐵) ↔ 𝐴𝐵))
2321, 22sylan9bb 738 . . . . 5 (((card‘𝐴) ≈ 𝐴 ∧ (card‘𝐵) ≈ 𝐵) → ((card‘𝐴) ≼ (card‘𝐵) ↔ 𝐴𝐵))
2413, 20, 23syl2anc 696 . . . 4 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → ((card‘𝐴) ≼ (card‘𝐵) ↔ 𝐴𝐵))
2511, 24mpbid 222 . . 3 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ ((card‘𝐴) ≠ ∅ ∧ (card‘𝐴) ⊆ (card‘𝐵))) → 𝐴𝐵)
2625expr 644 . 2 (((𝐴 ∈ dom card ∧ 𝐵𝑉) ∧ (card‘𝐴) ≠ ∅) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴𝐵))
277, 26pm2.61dane 3019 1 ((𝐴 ∈ dom card ∧ 𝐵𝑉) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139   ≠ wne 2932  Vcvv 3340   ⊆ wss 3715  ∅c0 4058   class class class wbr 4804  dom cdm 5266  ‘cfv 6049   ≈ cen 8118   ≼ cdom 8119  cardccrd 8951 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-8 2141  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-pow 4992  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  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-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-er 7911  df-en 8122  df-dom 8123  df-card 8955 This theorem is referenced by:  carddom2  8993
 Copyright terms: Public domain W3C validator