Theorem carden2a 9987

Description: If two sets have equal nonzero cardinalities, then they are equinumerous. This assertion and carden2b 9988 are meant to replace carden 10572 in ZF without AC. (Contributed by Mario Carneiro, 9-Jan-2013.)

Assertion

Ref	Expression
carden2a	⊢ (((card‘𝐴) = (card‘𝐵) ∧ (card‘𝐴) ≠ ∅) → 𝐴 ≈ 𝐵)

Proof of Theorem carden2a

Step	Hyp	Ref	Expression
1		df-ne 2931	. 2 ⊢ ((card‘𝐴) ≠ ∅ ↔ ¬ (card‘𝐴) = ∅)
2		ndmfv 6925	. . . . . . 7 ⊢ (¬ 𝐵 ∈ dom card → (card‘𝐵) = ∅)
3		eqeq1 2729	. . . . . . 7 ⊢ ((card‘𝐴) = (card‘𝐵) → ((card‘𝐴) = ∅ ↔ (card‘𝐵) = ∅))
4	2, 3	imbitrrid 245	. . . . . 6 ⊢ ((card‘𝐴) = (card‘𝐵) → (¬ 𝐵 ∈ dom card → (card‘𝐴) = ∅))
5	4	con1d 145	. . . . 5 ⊢ ((card‘𝐴) = (card‘𝐵) → (¬ (card‘𝐴) = ∅ → 𝐵 ∈ dom card))
6	5	imp 405	. . . 4 ⊢ (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → 𝐵 ∈ dom card)
7		cardid2 9974	. . . 4 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
8	6, 7	syl 17	. . 3 ⊢ (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → (card‘𝐵) ≈ 𝐵)
9		breq2 5145	. . . . 5 ⊢ ((card‘𝐴) = (card‘𝐵) → (𝐴 ≈ (card‘𝐴) ↔ 𝐴 ≈ (card‘𝐵)))
10		entr 9023	. . . . . 6 ⊢ ((𝐴 ≈ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝐴 ≈ 𝐵)
11	10	ex 411	. . . . 5 ⊢ (𝐴 ≈ (card‘𝐵) → ((card‘𝐵) ≈ 𝐵 → 𝐴 ≈ 𝐵))
12	9, 11	biimtrdi 252	. . . 4 ⊢ ((card‘𝐴) = (card‘𝐵) → (𝐴 ≈ (card‘𝐴) → ((card‘𝐵) ≈ 𝐵 → 𝐴 ≈ 𝐵)))
13		cardid2 9974	. . . . . 6 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
14		ndmfv 6925	. . . . . 6 ⊢ (¬ 𝐴 ∈ dom card → (card‘𝐴) = ∅)
15	13, 14	nsyl4 158	. . . . 5 ⊢ (¬ (card‘𝐴) = ∅ → (card‘𝐴) ≈ 𝐴)
16	15	ensymd 9022	. . . 4 ⊢ (¬ (card‘𝐴) = ∅ → 𝐴 ≈ (card‘𝐴))
17	12, 16	impel 504	. . 3 ⊢ (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → ((card‘𝐵) ≈ 𝐵 → 𝐴 ≈ 𝐵))
18	8, 17	mpd 15	. 2 ⊢ (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → 𝐴 ≈ 𝐵)
19	1, 18	sylan2b 592	1 ⊢ (((card‘𝐴) = (card‘𝐵) ∧ (card‘𝐴) ≠ ∅) → 𝐴 ≈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ≠ wne 2930  ∅c0 4316   class class class wbr 5141  dom cdm 5670  ‘cfv 6541   ≈ cen 8957  cardccrd 9956

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5292  ax-nul 5299  ax-pow 5357  ax-pr 5421  ax-un 7736

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3958  df-nul 4317  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4943  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-ord 6365  df-on 6366  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-er 8721  df-en 8961  df-card 9960

This theorem is referenced by:  card1  9989