Theorem carden2a 10006

Description: If two sets have equal nonzero cardinalities, then they are equinumerous. This assertion and carden2b 10007 are meant to replace carden 10591 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 2941	. 2 ⊢ ((card‘𝐴) ≠ ∅ ↔ ¬ (card‘𝐴) = ∅)
2		ndmfv 6941	. . . . . . 7 ⊢ (¬ 𝐵 ∈ dom card → (card‘𝐵) = ∅)
3		eqeq1 2741	. . . . . . 7 ⊢ ((card‘𝐴) = (card‘𝐵) → ((card‘𝐴) = ∅ ↔ (card‘𝐵) = ∅))
4	2, 3	imbitrrid 246	. . . . . 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 406	. . . 4 ⊢ (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → 𝐵 ∈ dom card)
7		cardid2 9993	. . . 4 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
8	6, 7	syl 17	. . 3 ⊢ (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → (card‘𝐵) ≈ 𝐵)
9		breq2 5147	. . . . 5 ⊢ ((card‘𝐴) = (card‘𝐵) → (𝐴 ≈ (card‘𝐴) ↔ 𝐴 ≈ (card‘𝐵)))
10		entr 9046	. . . . . 6 ⊢ ((𝐴 ≈ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝐴 ≈ 𝐵)
11	10	ex 412	. . . . 5 ⊢ (𝐴 ≈ (card‘𝐵) → ((card‘𝐵) ≈ 𝐵 → 𝐴 ≈ 𝐵))
12	9, 11	biimtrdi 253	. . . 4 ⊢ ((card‘𝐴) = (card‘𝐵) → (𝐴 ≈ (card‘𝐴) → ((card‘𝐵) ≈ 𝐵 → 𝐴 ≈ 𝐵)))
13		cardid2 9993	. . . . . 6 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
14		ndmfv 6941	. . . . . 6 ⊢ (¬ 𝐴 ∈ dom card → (card‘𝐴) = ∅)
15	13, 14	nsyl4 158	. . . . 5 ⊢ (¬ (card‘𝐴) = ∅ → (card‘𝐴) ≈ 𝐴)
16	15	ensymd 9045	. . . 4 ⊢ (¬ (card‘𝐴) = ∅ → 𝐴 ≈ (card‘𝐴))
17	12, 16	impel 505	. . 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 594	1 ⊢ (((card‘𝐴) = (card‘𝐵) ∧ (card‘𝐴) ≠ ∅) → 𝐴 ≈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∅c0 4333   class class class wbr 5143  dom cdm 5685  ‘cfv 6561   ≈ cen 8982  cardccrd 9975

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-er 8745  df-en 8986  df-card 9979

This theorem is referenced by:  card1  10008