Theorem carden2a 9957

Description: If two sets have equal nonzero cardinalities, then they are equinumerous. This assertion and carden2b 9958 are meant to replace carden 10542 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 6923	. . . . . . 7 ⊢ (¬ 𝐵 ∈ dom card → (card‘𝐵) = ∅)
3		eqeq1 2736	. . . . . . 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 407	. . . 4 ⊢ (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → 𝐵 ∈ dom card)
7		cardid2 9944	. . . 4 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
8	6, 7	syl 17	. . 3 ⊢ (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → (card‘𝐵) ≈ 𝐵)
9		breq2 5151	. . . . 5 ⊢ ((card‘𝐴) = (card‘𝐵) → (𝐴 ≈ (card‘𝐴) ↔ 𝐴 ≈ (card‘𝐵)))
10		entr 8998	. . . . . 6 ⊢ ((𝐴 ≈ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝐴 ≈ 𝐵)
11	10	ex 413	. . . . 5 ⊢ (𝐴 ≈ (card‘𝐵) → ((card‘𝐵) ≈ 𝐵 → 𝐴 ≈ 𝐵))
12	9, 11	syl6bi 252	. . . 4 ⊢ ((card‘𝐴) = (card‘𝐵) → (𝐴 ≈ (card‘𝐴) → ((card‘𝐵) ≈ 𝐵 → 𝐴 ≈ 𝐵)))
13		cardid2 9944	. . . . . 6 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
14		ndmfv 6923	. . . . . 6 ⊢ (¬ 𝐴 ∈ dom card → (card‘𝐴) = ∅)
15	13, 14	nsyl4 158	. . . . 5 ⊢ (¬ (card‘𝐴) = ∅ → (card‘𝐴) ≈ 𝐴)
16	15	ensymd 8997	. . . 4 ⊢ (¬ (card‘𝐴) = ∅ → 𝐴 ≈ (card‘𝐴))
17	12, 16	impel 506	. . 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 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∅c0 4321   class class class wbr 5147  dom cdm 5675  ‘cfv 6540   ≈ cen 8932  cardccrd 9926

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-er 8699  df-en 8936  df-card 9930

This theorem is referenced by:  card1  9959