Theorem ecexr 6606

Description: An inhabited equivalence class implies the representative is a set. (Contributed by Mario Carneiro, 9-Jul-2014.)

Assertion

Ref	Expression
ecexr	⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐵 ∈ V)

Proof of Theorem ecexr

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elimag 5014	. . . . 5 ⊢ (𝐴 ∈ (𝑅 “ {𝐵}) → (𝐴 ∈ (𝑅 “ {𝐵}) ↔ ∃𝑥 ∈ {𝐵}𝑥𝑅𝐴))
2	1	ibi 176	. . . 4 ⊢ (𝐴 ∈ (𝑅 “ {𝐵}) → ∃𝑥 ∈ {𝐵}𝑥𝑅𝐴)
3		df-ec 6603	. . . 4 ⊢ [𝐵]𝑅 = (𝑅 “ {𝐵})
4	2, 3	eleq2s 2291	. . 3 ⊢ (𝐴 ∈ [𝐵]𝑅 → ∃𝑥 ∈ {𝐵}𝑥𝑅𝐴)
5		df-rex 2481	. . . 4 ⊢ (∃𝑥 ∈ {𝐵}𝑥𝑅𝐴 ↔ ∃𝑥(𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴))
6		simpl 109	. . . . . 6 ⊢ ((𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴) → 𝑥 ∈ {𝐵})
7		velsn 3640	. . . . . 6 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
8	6, 7	sylib 122	. . . . 5 ⊢ ((𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴) → 𝑥 = 𝐵)
9	8	eximi 1614	. . . 4 ⊢ (∃𝑥(𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴) → ∃𝑥 𝑥 = 𝐵)
10	5, 9	sylbi 121	. . 3 ⊢ (∃𝑥 ∈ {𝐵}𝑥𝑅𝐴 → ∃𝑥 𝑥 = 𝐵)
11	4, 10	syl 14	. 2 ⊢ (𝐴 ∈ [𝐵]𝑅 → ∃𝑥 𝑥 = 𝐵)
12		isset 2769	. 2 ⊢ (𝐵 ∈ V ↔ ∃𝑥 𝑥 = 𝐵)
13	11, 12	sylibr 134	1 ⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐵 ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364  ∃wex 1506   ∈ wcel 2167  ∃wrex 2476  Vcvv 2763  {csn 3623   class class class wbr 4034   “ cima 4667  [cec 6599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-xp 4670  df-cnv 4672  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-ec 6603

This theorem is referenced by:  relelec  6643  ecdmn0m  6645